function - relation between a set of inputs and permissible outputs

Formula: ƒ(x)

$100 fee plus $30 per month. Write an expression that describes the cost of a gym membership after m

$100 fee plus $30 per month. Write an expression that describes the cost of a gym membership after m months.
Set up the cost function C(m) where m is the number of months you rent:
C(m) = Monthly membership fee * m + initial fee
[B]C(m) = 30m + 100[/B]

$45 and you add $2.25 each day

$45 and you add $2.25 each day
Let d be the number of days. Our Cost function C(d) is:
[B]C(d) = 2.25d + 45[/B]

(2,3)(4,5)(6,7)(8,9) represents a function

(2,3)(4,5)(6,7)(8,9) represents a function
Domain is the x-values:
x = (2, 4, 6, 8)
Range is the y-values:
y = (3, 5, 7, 9)
The function y, or f(x) is:
y = x + 1 where x = (2, 4, 6, 8)
Test this function for x = 2:
y = 2 + 1
y = 3
Test this function for x = 4:
y = 4 + 1
y = 5
Test this function for x = 6:
y = 6 + 1
y = 7
Test this function for x = 8:
y = 8 + 1
y = 9

1, 1/2, 1/4, 1/8, 1/16 The next number in the sequence is 1/32. What is the function machine you wou

1, 1/2, 1/4, 1/8, 1/16
The next number in the sequence is 1/32. What is the function machine you would use to find the nth term of this sequence?
Hint: look at the denominators
We notice that
1/2^0 = 1/1 = 1
1/2^1 = 1/2
1/2^2 = 1/4
1/2^3 = 1/8
1/2^4 = 1/32
So we write our explicit formula for term n:
f(n) = [B]1/2^(n - 1)[/B]

1, 8, 27, 64 What is the 10th term?

1, 8, 27, 64
What is the 10th term?
We see the following pattern:
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
We build our sequence function using this pattern:
f(n) = n^3
With n = 10, we have:
f(10) = 10^3
f(10) = [B]1,000[/B]

1/2, 3, 5&1/2, 8......203 What term is the number 203?

1/2, 3, 5&1/2, 8......203
What term is the number 203?
We see the following pattern:
1/2 = 2.5*1 - 2
3 = 2.5*2 - 2
5&1/2 = 2.5*3 - 2
8 = 2.5*4 - 2
We build our function
f(n) = 2.5n - 2
Set 2.5n - 2 = 203
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=2.5n-2%3D203&pl=Solve']equation solver[/URL], we get:
n = [B]82[/B]

100, 75, 50, 25, 0, -25 What is the next number? What is the 100th term?

100, 75, 50, 25, 0, -25 What is the next number? What is the 100th term?
Using point slope, we get (1, 100)(2, 75)
Our [URL='https://www.mathcelebrity.com/search.php?q=%281%2C+100%29%282%2C+75%29&x=0&y=0']series function becomes[/URL]
f(n) = -25n + 125
The next term is the 7th term:
f(7) = -25(7) + 125
f(7) = -175 + 125
f(7) = [B]-50
[/B]
The 100th term is found by n = 100:
f(100) = -25(100) + 125
f(100) = -2500 + 125
f(100) = [B]-2375[/B]

3 per ride r plus $10 to get into the park

3 per ride r plus $10 to get into the park
Cost function C(r) where r is rides:
C(r) = Rate per ride * number of rides + admission cost
[B]C(r) = 3r + 10[/B]

3, 6, 12, 24, 48 What is the function machine for this sequence?

3, 6, 12, 24, 48
What is the function machine for this sequence?
We see the following pattern:
3 * 2^0 = 3
3 * 2^1 = 6
3 * 2^2 = 12
3 * 2^3 = 24
3 * 2^4 = 48
Our function machine for term n is:
[B]f(n) = 3 * 2^(n - 1)[/B]

5, 14, 23, 32, 41....1895 What term is the number 1895?

5, 14, 23, 32, 41....1895 What term is the number 1895?
Set up a point slope for the first 2 points:
(1, 5)(2, 14)
Using [URL='https://www.mathcelebrity.com/search.php?q=%281%2C+5%29%282%2C+14%29&x=0&y=0']point slope formula, our series function[/URL] is:
f(n) = 9n - 4
To find what term 1895 is, we set 9n - 4 = 1895 and solve for n:
9n - 4 = 1895
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=9n-4%3D1895&pl=Solve']equation solver[/URL], we get:
n = [B]211[/B]

7, 10, 15, 22 What is the next number in the sequence? What is the 500th term?

7, 10, 15, 22
What is the next number in the sequence?
What is the 500th term?
We see that:
1^2 + 6 = 7
2^2 + 6 = 10
3^3 + 6 = 15
4^2 + 6 = 22
We build our function as f(n) = n^2 + 6
Next term in the sequence is f(5)
f(5) = 5^2 + 6
f(5) = 25 + 6
f(5) = [B]31
[/B]
Calculate the 500th term:
f(500) = 500^2 + 6
f(500) = 250,000 + 6
f(500) = [B]250,006[/B]

9, 3, 1, 1/3, 1/9 What is the next number in this sequence? What is the function machine for this se

9, 3, 1, 1/3, 1/9
What is the next number in this sequence?
What is the function machine for this sequence?
We see the following pattern in this sequence:
9 = 9/3^0
3 = 9/3^1
1 = 9/3^2
1/3 = 9/3^3
1/9 = 9/3^4
Our function machine formula is:
[B]f(n) = 9/3^(n - 1)
[/B]
Next term is the 6th term:
f(6) = 9/3^(6 - 1)
f(6) = 9/3^5
f(6) = 9/243
f(6) = [B]1/27[/B]

A $1,000 investment takes a 10% loss each year. What will be the value 3 years?

A $1,000 investment takes a 10% loss each year. What will be the value 3 years?
10% is 0.1. Our Balance function B(y) where y is the number of years since the start is:
B(y) = 1000(1 - 0.1)^y
B(y) = 1000(0.9)^y
We want to know B(3):
B(3) = 1000(0.9)^3
B(3) = 1000(0.729)
B(3) = [B]729[/B]

A 1975 comic book has appreciated 8% per year and originally sold for $0.26. What will the comic boo

A 1975 comic book has appreciated 8% per year and originally sold for $0.26. What will the comic book be worth in 2020
Calculate the number of years:
2020 - 1975 = 45
Set up the accumulation function A(t) where t is the number of years since 1975:
A(t) = 0.26(1.08)^t
We want A(45)
A(45) = 0.26(1.08)^45
A(45) = 0.26 * 32.9045
A(45) = [B]8.30[/B]

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to keep its daily costs at or below $500 per day. Which inequality shows the maximum number of pastries, p, that can be baked each day.
Set up the cost function C(p), where p is the number of pastries:
C(p) = Variable Cost + Fixed Cost
C(p) = 2.25p + 119.75
The problem asks for C(p) at or below $500 per day. The phrase [I]at or below[/I] means less than or equal to (<=).
[B]2.25p + 119.75 <= 500[/B]

A baseball card that was valued at $100 in 1970 has increased in value by 8% each year. Write a func

A baseball card that was valued at $100 in 1970 has increased in value by 8% each year. Write a function to model the situation the value of the card in 2020.Let x be number of years since 1970
The formula for accumulated value of something with a percentage growth p and years x is:
V(x) = Initial Value * (1 + p/100)^x
Set up our growth equation where 8% = 0.08 and V(y) for the value at time x and x = 2020 - 1970 = 50, we have:
V(x) = 100 * (1 + 8/100)^50
V(x) = 100 * (1.08)^50
V(x) = 100 * 46.9016125132
V(x) = [B]4690.16[/B]

A bicycle store costs $2750 per month to operate. The store pays an average of $45 per bike. The a

A bicycle store costs $2750 per month to operate. The store pays an average of $45 per bike. The average selling price of each bicycle is $95. How many bicycles must the store sell each month to break even?
Let the number of bikes be b.
Set up our cost function, where it costs $45 per bike to produce
C(b) = 45b
Set up our revenue function, where we earn $95 per sale for each bike:
R(b) = 95b
Set up our profit function, which is how much we keep after a sale:
P(b) = R(b) - C(b)
P(b) = 95b - 45b
P(b) = 50b
The problem wants to know how many bikes we need to sell to break-even. Note: break-even means profit equals operating cost, which in this case, is $2,750. So we set our profit function of 50b equal to $2,750
50b = 2750
[URL='https://www.mathcelebrity.com/1unk.php?num=50b%3D2750&pl=Solve']We type this equation into our search engine[/URL], and we get:
b = [B]55[/B]

a bicycle store costs $3600 per month to operate. The store pays an average of $60 per bike. the ave

a bicycle store costs $3600 per month to operate. The store pays an average of $60 per bike. the average selling price of each bicycle is $100. how many bicycles must the store sell each month to break even?
Cost function C(b) where b is the number of bikes:
C(b) = Variable Cost + Fixed Cost
C(b) = Cost per bike * b + operating cost
C(b) = 60b + 3600
Revenue function R(b) where b is the number of bikes:
R(b) = Sale price * b
R(b) = 100b
Break Even is when Cost equals Revenue, so we set C(b) = R(b):
60b + 3600 = 100b
To solve this equation for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=60b%2B3600%3D100b&pl=Solve']type it in our math engine[/URL] and we get:
b = [B]90[/B]

A boat costs 14950 and decrease in value by 7% per year how much will the boat be worth after 8 yea

A boat costs 14950 and decrease in value by 7% per year how much will the boat be worth after 8 years?
If a boat decreases in value 7% in value, then our new value each year is 100% - 7% = 93%. So we have a B(y) function where B(y) is the value of the boat after y years:
B(y) = 14,950 * (1 - 0.07)^y
Simplifying, we get:
B(y) = 14,950 * (0.93)^y
The problem asks for B(8)
B(8) = 14,950 * (0.93)^7
B(8) = 14,950 * 0..6017
B(8) = [B]8,995.43[/B]

A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books

A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books they make sell for $40 each.
[B][U]Set up Cost Function C(b) where b is the number of books:[/U][/B]
C(b) = Fixed Cost + Variable Cost x Number of Units
C(b) = 180,000 + 25(b)
[B]Set up Revenue Function R(b):[/B]
R(b) = 40b
Set them equal to each other
180,000 + 25b = 40b
Subtract 25b from each side:
15b = 180,000
Divide each side by 15
[B]b = 12,000 for break even[/B]

A bunny population is doubling every 2 years. There are currently 45 bunnies. How many will there be

A bunny population is doubling every 2 years. There are currently 45 bunnies. How many will there be in 10 years?
Find the number of doubling periods:
Number of Doubling periods = Time / Doubling period
Number of Doubling periods = 10/2
Number of Doubling periods = 5
Create a function to determine the amount of bunnies after each doubling period:
B(n) = 45 * 2^n
Since we calculated 5 doubling periods, we want B(5):
B(5) = 45 * 2^5
B(5) = 45 * 32
B(5) = [B]1,440[/B]

A cab charges $5 for the ride plus $1.25 per mile. How much will a 53 mile trip cost?

A cab charges $5 for the ride plus $1.25 per mile. How much will a 53 mile trip cost?
We set up our cost function C(m) where m is the number of miles:
C(m) = 1.25m + 5
The problem asks for C(53):
C(53) = 1.25(53) + 5
C(53) = 66.25 + 5
C(53) = [B]$71.25[/B]

A cab company charges $5 per cab ride, plus an additional $1 per mile driven , How long is a cab rid

A cab company charges $5 per cab ride, plus an additional $1 per mile driven , How long is a cab ride that costs $13?
Let the number of miles driven be m. Our cost function C(m) is:
C(m) = Cost per mile * m + cab cost
C(m) = 1m + 5
The problem asks for m when C(m) = 13:
1m + 5 = 13
To solve this equation for m, [URL='https://www.mathcelebrity.com/1unk.php?num=1m%2B5%3D13&pl=Solve']we type it in our search engine[/URL] and we get:
m = [B]8[/B]

A cable company charges $75 for installation plus $20 per month. Another cable company offers free i

A cable company charges $75 for installation plus $20 per month. Another cable company offers free installation but charges $35 per month. For how many months of cable service would the total cost from either company be the same
[U]Set ups the cost function for the first cable company C(m) where m is the number of months:[/U]
C(m) = cost per month * m + installation fee
C(m) = 20m + 75
[U]Set ups the cost function for the second cable company C(m) where m is the number of months:[/U]
C(m) = cost per month * m + installation fee
C(m) = 35m
The problem asks for m when both C(m) functions are equal. So we set both C(m) functions equal and solve for m:
20m + 75 = 35m
To solve for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=20m%2B75%3D35m&pl=Solve']type this equation into our search engine[/URL] and we get:
m = [B]5[/B]

A car drives 3 feet the first second, 9 feet in the next second, and 27 feet in the third second. If

A car drives 3 feet the first second, 9 feet in the next second, and 27 feet in the third second. If the pattern stays the same, how far will the car have traveled after 5 seconds, in feet?
Our pattern is found by the distance function D(t), where we have 3 to the power of the time (t) in seconds as seen below:
D(t) = 3^t
The problem asks for D(5):
D(5) = 3^5
[URL='https://www.mathcelebrity.com/powersq.php?sqconst=+6&num=3%5E5&pl=Calculate']D(5)[/URL] = [B]243[/B]

A car is purchased for $19000. After each year, the resale value decreases by 30% . What will the re

A car is purchased for $19000. After each year, the resale value decreases by 30% . What will the resale value be after 4 years?
Set up a book value function B(t) where t is the number of years after purchase date. If an asset decreases by 30%, we subtract it from the original 100% of the starting value at time t:
B(t) = 19,000(1-0.3)^t
Simplifying this, we get:
B(t) = 19,000(0.7)^t <-- I[I]f an asset decreases by 30%, it keeps 70% of it's value from the prior period[/I]
The problem asks for B(4):
B(4) = 19,000(0.7)^4
B(4) = 19,000(0.2401)
B(4) = [B]4,561.90[/B]

A car rents $35 per day plus 15 cents per mile driven

A car rents $35 per day plus 15 cents per mile driven
Set up the cost function C(m) where m is the number of miles driven:
C(m) = Cost per mile * m + Daily Fee
[B]C(m) = 0.15m + 35[/B]

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a person have to play to spend at least $40?
Let g be the number of games. The Spend function S(g) is:
S(g) = Cost per game * number of games + admission price
S(g) = 4g + 15
The problem asks for g when S(g) is at least 40. At least is an inequality using the >= sign:
4g + 15 >= 40
To solve this inequality for g, we type it in our search engine and we get:
g >= 6.25
Since you can't play a partial game, we round up and get:
[B]g >= 7[/B]

A car’s purchase price is $24,000. At the end of each year, the value of the car is three-quarters o

A car’s purchase price is $24,000. At the end of each year, the value of the car is three-quarters of the value at the beginning of the year. Write the first four terms of the sequence of the car’s value at the end of each year.
three-quarters means 3/4 or 0.75. So we have the following function P(y) where y is the number of years since purchase price:
P(y) = 24000 * 0.75^y
First four terms:
P(1) = 24000 * 0.75 = [B]18000[/B]
P(2) = 18000 * 0.75 = [B]13500[/B]
P(3) = 13500 * 0.75 = [B]10125[/B]
P(4) = 10125 * 0.75 = [B]7593.75[/B]

A cell phone costs $20 for 400 minutes and $2 for each extra minute. Gina uses 408 minutes. How m

A cell phone costs $20 for 400 minutes and $2 for each extra minute. Gina uses 408 minutes. How much will it cost?
Set up the cost function for minutes (m) if m is greater than or equal to 400
C(m) = 20 + 2(m - 400)
For m = 408, we have:
C(408) = 20 + 2(408 - 400)
C(408) = 20 + 2(8)
C(408) = [B]36[/B]

A cell phone plan charges $1.25 for the first 400 minutes and $0.25 for each additional minute, x. W

A cell phone plan charges $1.25 for the first 400 minutes and $0.25 for each additional minute, x. Which represents the cost of the cell phone plan?
Let C(x) be the cost function where x is the number of minutes we have:
[B]C(x) = 1.25(min(400, x)) + 0.25(Max(0, 400 - x))[/B]

A cell phone plan costs $20 a month and includes 200 free minutes. Each additional minute costs 5 ce

A cell phone plan costs $20 a month and includes 200 free minutes. Each additional minute costs 5 cents. If you use your cell phone for at least 200 minutes a month, write a function C(x) that represents the total cost per x minutes.
We add the flat rate per month to 5% of the number of minutes [U]over[/U] 200:
[B]C(x) = 20 + 0.05(x - 200)[/B]

A cheetah travels at a rate of 90 feet per second. The distance d traveled by the cheetah is a func

A cheetah travels at a rate of 90 feet per second. The distance d traveled by the cheetah is a function of seconds traveled t. Write a rule for the function. How far will the cheetah travel in 25 seconds?
Distance, or D(t) is expressed as a function of rate and time below:
Distance = Rate x Time
For the cheetah, we have D(t) as:
D(t) = 90ft/sec(t)
The problem asks for D(25):
D(25) = 90(25)
D(25) = [B]2,250 feet[/B]

A city has a population of 240,000 people. Suppose that each year the population grows by 7.25%. Wha

A city has a population of 240,000 people. Suppose that each year the population grows by 7.25%. What will the population be after 9 years?
Let's build a population function P(t), where t is the number of years since right now.
P(t) = 240,000(1.0725)^t <-- 7.25% as a decimal is 0.0725
The question asks for P(9)
P(9) = 240,000(1.0725)^9
P(9) = 240,000(1.87748)
P(9) = [B]450,596[/B]

A city has a population of 240,000 people. Suppose that each year the population grows by 8%. What w

A city has a population of 240,000 people. Suppose that each year the population grows by 8%. What will the population be after 5 years?
[U]Set up our population function[/U]
P(t) = 240,000(1 + t)^n where t is population growth rate percent and n is the time in years
[U]Evaluate at t = 0.08 and n = 5[/U]
P(5) = 240,000(1 + 0.08)^5
P(5) = 240,000(1.08)^5
P(5) = 240,000 * 1.4693280768
[B]P(5) = 352638.73 ~ 352,639[/B]

a collection of 7 pencils, every week 3 more pencils are added How many weeks will it take to have 3

a collection of 7 pencils, every week 3 more pencils are added How many weeks will it take to have 30 pencils?
Set up a function, P(w), where w is the number of weeks, and P(w) is the total amount of pencils after w weeks. We have:
P(w) = 3w + 7
We want to know what w is when P(w) = 30
3w + 7 = 30
[URL='https://www.mathcelebrity.com/1unk.php?num=3w%2B7%3D30&pl=Solve']Typing this equation into our search engine[/URL], we get:
w = 7.6667
We round up to the nearest integer, so we get [B]w = 8[/B]

A company has a fixed cost of $26,000 / month when it is producing printed tapestries. Each item tha

A company has a fixed cost of $26,000 / month when it is producing printed tapestries. Each item that it makes has its own cost of $34. One month the company filled an order for 2400 of its tapestries, selling each item for $63. How much profit was generated by the order?
[U]Set up Cost function C(t) where t is the number of tapestries:[/U]
C(t) = Cost per tapestry * number of tapestries + Fixed Cost
C(t) = 34t + 26000
[U]Set up Revenue function R(t) where t is the number of tapestries:[/U]
R(t) = Sale Price * number of tapestries
R(t) = 63t
[U]Set up Profit function P(t) where t is the number of tapestries:[/U]
P(t) = R(t) - C(t)
P(t) = 63t - (34t + 26000)
P(t) = 63t - 34t - 26000
P(t) = 29t - 26000
[U]The problem asks for profit when t = 2400:[/U]
P(2400) = 29(2400) - 26000
P(2400) = 69,600 - 26,000
P(2400) = [B]43,600[/B]

A company has a fixed cost of $34,000 and a production cost of $6 for each unit it manufactures. A u

A company has a fixed cost of $34,000 and a production cost of $6 for each unit it manufactures. A unit sells for $15
Set up the cost function C(u) where u is the number of units is:
C(u) = Cost per unit * u + Fixed Cost
C(u) = [B]6u + 34000[/B]
Set up the revenue function R(u) where u is the number of units is:
R(u) = Sale price per unit * u
R(u) = [B]15u[/B]

A company is planning to manufacture a certain product. The fixed costs will be $474778 and it will

A company is planning to manufacture a certain product. The fixed costs will be $474778 and it will cost $293 to produce each product. Each will be sold for $820. Find a linear function for the profit, P , in terms of units sold, x .
[U]Set up the cost function C(x):[/U]
C(x) = Cost per product * x + Fixed Costs
C(x) = 293x + 474778
[U]Set up the Revenue function R(x):[/U]
R(x) = Sale Price * x
R(x) = 820x
[U]Set up the Profit Function P(x):[/U]
P(x) = Revenue - Cost
P(x) = R(x) - C(x)
P(x) = 820x - (293x + 474778)
P(x) = 820x - 293x - 474778
[B]P(x) = 527x - 474778[/B]

A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat.

A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
[U]Set up Cost function C(b) where t is the number of tapestries:[/U]
C(b) = Cost per boat * number of boats + Fixed Cost
C(b) = 50b + 1500
[U]Set up Revenue function R(b) where t is the number of tapestries:[/U]
R(b) = Sale Price * number of boats
R(b) = 75b
[U]Break even is where Revenue equals Cost, or Revenue minus Cost is 0, so we have:[/U]
R(b) - C(b) = 0
75b - (50b + 1500) = 0
75b - 50b - 1500 = 0
25b - 1500 = 0
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=25b-1500%3D0&pl=Solve']type this equation in our math engine[/URL] and we get:
b = [B]60[/B]

A company specializes in personalized team uniforms. It costs the company $15 to make each uniform a

A company specializes in personalized team uniforms. It costs the company $15 to make each uniform along with their fixed costs at $640. The company plans to sell each uniform for $55.
[U]The cost function for "u" uniforms C(u) is given by:[/U]
C(u) = Cost per uniform * u + Fixed Costs
[B]C(u) = 15u + 640[/B]
Build the revenue function R(u) where u is the number of uniforms:
R(u) = Sale Price per uniform * u
[B]R(u) = 55u[/B]
Calculate break even function:
Break even is where Revenue equals cost
C(u) = R(u)
15u + 640 = 55u
To solve for u, we [URL='https://www.mathcelebrity.com/1unk.php?num=15u%2B640%3D55u&pl=Solve']type this equation into our search engine[/URL] and we get:
u = [B]16
So we break even selling 16 uniforms[/B]

A company that manufactures lamps has a fixed monthly cost of $1800. It costs $90 to produce each l

A company that manufactures lamps has a fixed monthly cost of $1800. It costs $90 to produce each lamp, and the selling price is $150 per lamp.
Set up the Cost Equation C(l) where l is the price of each lamp:
C(l) = Variable Cost x l + Fixed Cost
C(l) = 90l + 1800
Determine the revenue function R(l)
R(l) = 150l
Determine the profit function P(l)
Profit = Revenue - Cost
P(l) = 150l - (90l + 1800)
P(l) = 150l - 90l - 1800
[B]P(l) = 60l - 1800[/B]
Determine the break even point:
Breakeven --> R(l) = C(l)
150l = 90l + 1800
[URL='https://www.mathcelebrity.com/1unk.php?num=150l%3D90l%2B1800&pl=Solve']Type this into the search engine[/URL], and we get [B]l = 30[/B]

A companys cost function is C(x) = 16x2 + 900 dollars, where x is the number of units. Find th

A companys cost function is C(x) = 16x^2 + 900 dollars, where x is the number of units. Find the marginal cost function.
Marginal Cost is the derivative of the Cost function.
[B]C'(x) = 32x[/B]

A corn refining company produces corn gluten cattle feed at a variable cost of $84 per ton. If fixe

A corn refining company produces corn gluten cattle feed at a variable cost of $84 per ton. If fixed costs are $110,000 per month and the feed sells for $132 per ton, how many tons should be sold each month to have a monthly profit of $560,000?
[U]Set up the cost function C(t) where t is the number of tons of cattle feed:[/U]
C(t) = Variable Cost * t + Fixed Costs
C(t) = 84t + 110000
[U]Set up the revenue function R(t) where t is the number of tons of cattle feed:[/U]
R(t) = Sale Price * t
R(t) = 132t
[U]Set up the profit function P(t) where t is the number of tons of cattle feed:[/U]
P(t) = R(t) - C(t)
P(t) = 132t - (84t + 110000)
P(t) = 132t - 84t - 110000
P(t) = 48t - 110000
[U]The question asks for how many tons (t) need to be sold each month to have a monthly profit of 560,000. So we set P(t) = 560000:[/U]
48t - 110000 = 560000
[U]To solve for t, we [URL='https://www.mathcelebrity.com/1unk.php?num=48t-110000%3D560000&pl=Solve']type this equation into our search engine[/URL] and we get:[/U]
t =[B] 13,958.33
If the problem asks for whole numbers, we round up one ton to get 13,959[/B]

a family went to a baseball game. the cost to park the car was $5 AND THE COST PER TICKET WAS $21. W

a family went to a baseball game. the cost to park the car was $5 AND THE COST PER TICKET WAS $21. WRITE A LINEAR FUNCTION IN THE FORM Y=MX+B, FOR THE TOTAL COST OF GOING TO THE BASEBALL GAME,Y, AND THE TOTAL NUMBER PEOPLE IN THE FAMILY,X.
We have:
[B]y = 21x + 5[/B]
Since the cost of each ticket is $21, we multiply this by x, the total number of people in the family.
We add 5 as the cost to park the car, which fits the entire family, and is a one time cost.

A grandmother deposited $5,000 in an account that pays 8% per year compounded annually when her gran

A grandmother deposited $5,000 in an account that pays 8% per year compounded annually when her granddaughter was born. What will the value of the account be when the granddaughter reaches her 16th birthday?
We have the accumulation function A(t) = 5,000(1.08)^t.
For t = 16, we have:
A(16) = 5,000(1.08)^16
A(16) = 5,000*3.42594264333
A(16) = [B]17,129.71[/B]

A group of 4 adults and 5 children is visiting an amusement park. Admission is $15 per adult and $9

A group of 4 adults and 5 children is visiting an amusement park. Admission is $15 per adult and $9 per child. Find the total cost of admission for the group.
Set up the cost function for adults and children:
C(a, c) = 15a + 9c
We want the cost for 4 adults and 5 children
C(4, 5) = 15(4) + 9(5)
C(4, 5) = 60 + 45
C(4, 5) = [B]105[/B]

A heating company charges $60 per hour plus $54 for a service call. Let n be the number of hours t

A heating company charges $60 per hour plus $54 for a service call. Let n be the number of hours the technician works at your house.
The cost function C(n) where n is the number of hours is:
C(n) = Hourly Rate * hours + Service Call Charge
[B]C(n) = 60n + 54[/B]

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus $15 per hour. How long is a job for which companies will charge the same amount?
Set up the cost function C(h) where h is the number of hours.
Company 1:
C(h) = 12h + 376
Company 2:
C(h) = 15h + 280
To see when the companies charge the same amount, set both C(h) functions equal to each other.
12h + 376 = 15h + 280
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B376%3D15h%2B280&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]32[/B]

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus $15 per hour. How long is a job for which both companies will charge the same amount?
[U]Set up the cost function for the first company C(h) where h is the number of hours:[/U]
C(h) = Hourly Rate * h + flat rate
C(h) = 12h + 376
[U]Set up the cost function for the first company C(h) where h is the number of hours:[/U]
C(h) = Hourly Rate * h + flat rate
C(h) = 15h + 280
The problem asks how many hours will it take for both companies to charge the same. So we set the cost functions equal to each other:
12h + 376 = 15h + 280
Plugging this equation [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B376%3D15h%2B280&pl=Solve']into our search engine and solving for h[/URL], we get:
h = [B]32[/B]

A house valued at 70,000 in 1989 increased in value to 125,000 in 2000. Find a function which gives

A house valued at 70,000 in 1989 increased in value to 125,000 in 2000. Find a function which gives the value of the house, v, as a function of y, the number of years after 1989.
Let's determine the years:
2000 - 1989 = 11
Let's determine the change in value:
125,000 - 70,000 = 55,000
Assuming a linear progression, we have:
55,000/11 = 5,000 per year increase
[B]y = 70,000 + 5,000v[/B] where v is the number of years after 1989
Plug in 11 to check our work
y = 70,000 + 5,000(11)
y = 70,000 + 55,000
y = 125,000

A line in the xy-plane passes through the origin and has a slope of 4/5. What points lie on that lin

A line in the xy-plane passes through the origin and has a slope of 4/5. What points lie on that line.
Our line equation is:
y = mx + b
We're given:
m = 4/5
(x, y) = (0, 0)
So we have:
0 = 4/5(0) + b
0 = 0 + b
b = 0
Therefore, our line equation is:
y = 4/5x
[URL='https://www.mathcelebrity.com/function-calculator.php?num=y%3D4%2F5x&pl=Calculate']Start plugging in values here to get a list of points[/URL]

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus 7 cents per check. How many checks should be written each month to make the credit union a better deal?
Set up the cost function B(c) for the local bank where c is the number of checks:
B(c) = 0.03c + 19
Set up the cost function B(c) for the credit union where c is the number of checks:
B(c) = 0.07c + 7
We want to find out when:
0.07c + 7 < 0.03c + 19
[URL='https://www.mathcelebrity.com/1unk.php?num=0.07c%2B7%3C0.03c%2B19&pl=Solve']Typing this inequality into our search engine[/URL], we get:
c < 300

A mail courier charges a base fee of $4.95 plus $11.90 per package being delivered. If x represents

A mail courier charges a base fee of $4.95 plus $11.90 per package being delivered. If x represents the number of packages delivered, which of the following equations could be used to find y, the total cost of mailing packages?
Set up the cost function y = C(x)
[B]C(x) = 4.95 + 11.90x[/B]

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $10 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $10 for each unit produced. The product sells for $22/unit.
The cost function for each unit u is:
C(u) = Variable Cost * Units + Fixed Cost
C(u) = 10u + 100000
The revenue function R(u) is:
R(u) = 22u
We want the break-even point, which is where:
C(u) = R(u)
10u + 100000 = 22u
[URL='https://www.mathcelebrity.com/1unk.php?num=10u%2B100000%3D22u&pl=Solve']Typing this equation into our search engine[/URL], we get:
u =[B]8333.33[/B]

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $12 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $12 for each unit produced. The product sells for $20/unit
[U]Cost Function C(u) where u is the number of units:[/U]
C(u) = cost per unit * u + fixed cost
C(u) = 12u + 100000
[U]Revenue Function R(u) where u is the number of units:[/U]
R(u) = Sale price * u
R(u) = 20u
Break even point is where C(u) = R(u):
C(u) = R(u)
12u + 100000 = 20u
To solve for u, we [URL='https://www.mathcelebrity.com/1unk.php?num=12u%2B100000%3D20u&pl=Solve']type this equation into our search engine[/URL] and we get:
u = [B]12,500[/B]

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $14 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $14 for each unit produced. The product sells for $20/unit.
Let u be the number of units. We have a cost function C(u) as:
C(u) = Variable cost * u + Fixed Cost
C(u) = 14u + 100000
[U]We have a revenue function R(u) with u units as:[/U]
R(u) = Sale Price * u
R(u) = 20u
[U]We have a profit function P(u) with u units as:[/U]
Profit = Revenue - Cost
P(u) = R(u) - C(u)
P(u) = 20u - (14u + 100000)
P(u) = 20u - 14u - 100000
P(u) = 6u - 1000000

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produce

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produced. The product sells for $10/unit.
Set up cost function where u equals each unit produced:
C(u) = 7u + 25,500
Set up revenue function
R(u) = 10u
Break Even is where Cost equals Revenue
7u + 25,500 = 10u
Plug this into our [URL='http://www.mathcelebrity.com/1unk.php?num=7u%2B25500%3D10u&pl=Solve']equation calculator[/URL] to get [B]u = 8,500[/B]

A mechanic charges $45 per hour and parts cost $125. Write an expression for the total if the mechan

A mechanic charges $45 per hour and parts cost $125. Write an expression for the total if the mechanic works h hours.
Set up the cost function C(h) where h is the number of hours worked:
C(h) = Hourly Rate * h + parts
C(h) = [B]45h + 125[/B]

A mechanic charges $50 to inspect your heater, plus $80 per hour to work on it. You owe the mechani

A mechanic charges $50 to inspect your heater, plus $80 per hour to work on it. You owe the mechanic a total of $310. Write and solve an equation to find the amount of time h (in hours) the mechanic works on your heater.
We calculate the cost function C(h) as:
C(h) = Hourly Rate * hours + Flat Fee Inspection
C(h) = 80h + 50 <-- this is our cost equation
Now, we want to solve for h when C(h) = 310
80h + 50 = 310
[URL='https://www.mathcelebrity.com/1unk.php?num=80h%2B50%3D310&pl=Solve']We type this equation into our search engine[/URL] and we get:
h = [B]3.25[/B]

A mechanic will charge a new customer $45.00 for an initial diagnosis plus $20 an hour of labor. How

A mechanic will charge a new customer $45.00 for an initial diagnosis plus $20 an hour of labor. How long did the mechanic work on a car if he charged the customer $165?
We set up a cost function C(h) where h is the number of hours of labor:
C(h) = Hourly Labor Rate * h + Initial Diagnosis
C(h) = 20h + 45
The problem asks for the number of hours if C(h) = 165. So we set our cost function C(h) above equal to 165:
20h + 45 = 165
To solve for h, [URL='https://www.mathcelebrity.com/1unk.php?num=20h%2B45%3D165&pl=Solve']we plug this equation into our search engine[/URL] and we get:
h = [B]6[/B]

A monster energy drink has 164 mg of caffeine. Each hour your system reduces the amount of caffeine

A monster energy drink has 164 mg of caffeine. Each hour your system reduces the amount of caffeine by 12%. Write an equation that models the amount of caffeine that remains in your body after you drink an entire monster energy.
Set up a function C(h) where he is the number of hours after you drink the Monster energy drink:
Since 12% as a decimal is 0.12, we have:
C(h) = 164 * (1 - 0.12)^h <-- we subtract 12% since your body flushes it out
[B]C(h) = 164 * (0.88)^h[/B]

A music app charges $2 to download the app plus $1.29 per song download. Write and solve a linear eq

A music app charges $2 to download the app plus $1.29 per song download. Write and solve a linear equation to find the total cost to download 30 songs
Set up the cost function C(s) where s is the number of songs:
C(s) = cost per song * s + download fee
Plugging in our numbers for s = 30 and a download fee of $2 and s = 1.29, we have:
C(30) = 1.29(30) + 2
C(30) = 38.7 + 2
C(30) = [B]40.7[/B]

A music app charges $2 to download the app plus $1.29 per song downloaded

A music app charges $2 to download the app plus $1.29 per song downloaded
Let d be the number of downloads. The cost function C(d) is:
C(d) = cost per download * d + download fee
[B]C(d) = 1.29d + 2[/B]

A music app charges $2 to download the app plus $1.29 per song downloaded. Write and solve a linear

A music app charges $2 to download the app plus $1.29 per song downloaded. Write and solve a linear equation to find the total cost to download 30 songs.
Let the number of songs be s. And the cost function be C(s). We have:
C(s) = Price per song downloaded * s + app download charge
C(s) = 1.29s + 2
The problem asks for C(30):
C(3) = 1.29(30) + 2
C(3) = 38.7 +2
C(3) = $[B]40.7[/B]

A new car worth $24,000 is depreciating in value by $3,000 per year , how many years till the cars v

A new car worth $24,000 is depreciating in value by $3,000 per year , how many years till the cars value will be $9,000
We have a flat rate depreciation each year. Set up the function D(t) where t is the number of years of depreciation:
D(t) = 24000 - 3000t
The problem asks for the time (t) when D(t) = 9000. So we set D(t) = 9000
24000 - 3000 t = 9000
To solve for t, [URL='https://www.mathcelebrity.com/1unk.php?num=24000-3000t%3D9000&pl=Solve']we plug this function into our search engine[/URL] and we get:
t = [B]5[/B]

a package of soccer accessories costs $25 for cleats, $14 for shin guards , and $12 for a ball. Writ

a package of soccer accessories costs $25 for cleats, $14 for shin guards , and $12 for a ball. Write two equivalent expressions for the total cost of 9 accessory package. Then find the cost.
Let c be the number of cleats, s be the number of shin guards, and b be the number of balls. We have the following cost function for 9 accessory packages:
[B]9(25c + 14s + 12b)[/B]
But if we multiply through, we get an equivalent expression:
[B]225c + 126s + 108b[/B]

A peanut vendor has initial start up costs of $7600 and variable costs of $0.70 per bag of peanuts.

A peanut vendor has initial start up costs of $7600 and variable costs of $0.70 per bag of peanuts. What is the cost function?
We set up the cost function C(b) where b is the number of bags:
C(b) = Cost per bag * b + Start up costs
Plugging in our numbers, we get:
[B]C(b) = 0.70b + 7600[/B]

A person is earning 600 per day to do a certain job. Express the total salary as a function of the n

A person is earning 600 per day to do a certain job. Express the total salary as a function of the number of days that the person works.
Set up the salary function S(d) where d is the number of days that the person works:
S(d) = Daily Rate * d
[B]S(d) = 600d[/B]

A Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exi

A Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exist after 3 days?
Determine the amount of tripling periods:
[LIST]
[*]There are 24 hours in a day.
[*]24 hours in a day * 3 days = 72 hours
[*]72 hours / 6 hours tripling period = 12 tripling periods
[/LIST]
Our bacteria population function B(t) where t is the amount of tripling periods. Tripling means we multiply by 3, so we have:
B(t) = 2000 * 3^t
with t = 12 tripling periods, we have:
B(12) = 2000 * 3^12
B(12) = 2000 * 531441
B(12) = [B]1,062,882,000[/B]

A plant is 15 cm high and grows 4.5 cm every month. How many months will it take until the plant is

A plant is 15 cm high and grows 4.5 cm every month. How many months will it take until the plant is 27.5 cm
We set up the height function H(m) where m is the number of months since now. We have:
H(m) = 4.5m + 15
We want to know when H(m) = 27.5, so we set our H(m) function equal to 27.5:
4.5m + 15 = 27.5
To solve for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=4.5m%2B15%3D27.5&pl=Solve']type this equation into our search engine[/URL] and we get:
m = 2.78
So we round up to [B]3 whole months[/B]

A plumber charges $45 for a house call plus $25 for each hour worked.Let h represent the number of h

A plumber charges $45 for a house call plus $25 for each hour worked.Let h represent the number of hours worked. Write the expression that shows how much a plumber charges for a job. Then find how much the plumbers charges for a job lasting 4 hours
[U]Set up the cost function C(h) where h is the number of hours:[/U]
C(h) = Hours worked * hourly rate + house call fee
[B]C(h) = 25h + 45 <-- This is the expression for how much the plumber charges for a job
[/B]
[U]Now determine how much the plumber charges for a job lasting 4 hours[/U]
We want C(4)
C(4) = 25(4) + 45
C(4) = 100 + 45
C(4) = [B]$145[/B]

A plumber charges $50 to visit a house plus $40 for every hour of work.

A plumber charges $50 to visit a house plus $40 for every hour of work.
Set up the cost function in terms of hours (h) using the flat fee of $50
[B]C(h) = 40h + 50[/B]

A population of wolves on an island starts at 5 if the population doubles every 10 years, what will

A population of wolves on an island starts at 5 if the population doubles every 10 years, what will be the population in 90 years?
If the population doubles every 10 years, we have 90/10 = 9 doubling periods.
Our population function P(t) is where t is the doubling period
P(t) = 5(2^t)
The problem asks for P(9):
P(9) = 5(2^9)
P(9) = 5(512)
P(9) = [B]2,560[/B]

A pot of soup, currently 66°C above room temperature, is left out to cool. If that temperature diffe

A pot of soup, currently 66°C above room temperature, is left out to cool. If that temperature difference decreases by 10% per minute, then what will the difference be in 17 minutes?
We set up the temperature function T(m), where m is the number of minutes of cooling. With 10% = 0.1, we have:
T(m) = 66 * (1 - 0.10)^m
The problem asks for T(17) [U]and[/U] the difference temperature:
T(17) = 66 * 0.9^17
T(17) = 66 * 0.16677181699
T(17) = [B]11.01C[/B]
[B][/B]
[U]Calculate the difference in temperature[/U]
Difference = Starting Temperature - Ending Temperature
Difference = 66 - 11.01
Difference = 66 - 11.01 = [B]54.99 ~ 55[/B]

A pretzel factory has daily fixed costs of $1100. In addition, it costs 70 cents to produce each bag

A pretzel factory has daily fixed costs of $1100. In addition, it costs 70 cents to produce each bag of pretzels. A bag of pretzels sells for $1.80.
[U]Build the cost function C(b) where b is the number of bags of pretzels:[/U]
C(b) = Cost per bag * b + Fixed Costs
C(b) = 0.70b + 1100
[U]Build the revenue function R(b) where b is the number of bags of pretzels:[/U]
R(b) = Sale price * b
R(b) = 1.80b
[U]Build the revenue function P(b) where b is the number of bags of pretzels:[/U]
P(b) = Revenue - Cost
P(b) = R(b) - C(b)
P(b) = 1.80b - (0.70b + 1100)
P(b) = 1.80b = 0.70b - 1100
P(b) = 1.10b - 1100

A promotional deal for long distance phone service charges a $15 basic fee plus $0.05 per minute for

A promotional deal for long distance phone service charges a $15 basic fee plus $0.05 per minute for all calls. If Joe's phone bill was $60 under this promotional deal, how many minutes of phone calls did he make? Round to the nearest integer if necessary.
Let m be the number of minutes Joe used. We have a cost function of:
C(m) = 0.05m + 15
If C(m) = 60, then we have:
0.05m + 15 = 60
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05m%2B15%3D60&pl=Solve']Typing this equation into our search engine[/URL], we get:
m = [B]900[/B]

A rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation t

A rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation to determine the break even point?
Set up the cost functions for Rental Truck 1 (R1) and Rental Truck 2 (R2) where m is the number of miles
R1(m) = 0.59m + 49.95
R2(m) = 0.99m + 39.95
Break even is when we set the cost functions equal to one another:
0.59m + 49.95 = 0.99m + 39.95
[URL='https://www.mathcelebrity.com/1unk.php?num=0.59m%2B49.95%3D0.99m%2B39.95&pl=Solve']Typing this equation into the search engine[/URL], we get [B]m = 25[/B].

A repair bill for a car is $648.45. The parts cost $265.95. The labor cost is $85 per hour. Write an

A repair bill for a car is $648.45. The parts cost $265.95. The labor cost is $85 per hour. Write and solve an equation to find the number of hours spent repairing the car.
Let h be the number of hours spent repairing the car. We set up the cost function C(h):
C(h) = Labor Cost per hour * h + Parts Cost
We're given C(h) = 648.85, parts cost = 265.95, and labor cost per hour of 85, so we have:
85h + 265.95 = 648.85
To solve this equation, we [URL='https://www.mathcelebrity.com/1unk.php?num=85h%2B265.95%3D648.85&pl=Solve']type this into our search engine[/URL] and we get:
h = [B]4.5[/B]

A revenue function is R(x) = 22x and a cost function is C(x) = -9x + 341. The break-even point is

A revenue function is R(x) = 22x and a cost function is C(x) = -9x + 341. The break-even point is
Break even is when C(x) = R(x). So we set them equal and solve for x:
-9x + 341 = 22x
Typing[URL='https://www.mathcelebrity.com/1unk.php?num=-9x%2B341%3D22x&pl=Solve'] this equation into our search engine[/URL], we get:
x = [B]11[/B]

A salesperson drove 9 hours. How long will he have driven t hours later?

Set up a function where t is the number of hours driven, and f(t) is the distance driven after t hours:
[B]f(t) = 9t[/B]

A salesperson receives a base salary of $300 per week and a commission of 15% on all sales over $5,0

A salesperson receives a base salary of $300 per week and a commission of 15% on all sales over $5,000. If x represents the salesperson’s weekly sales, express the total weekly earnings E(x) as a function of x and simplify the expression. Then find E(2,000) and E(7,000) and E(10,000).
15% as a decimal is written as 0.15.
Build our weekly earnings function
E(x) = Commission + Base Salary
E(x) = 0.15(Max(0, x - 5000)) + 300
Now find the sales salary for 2,000, 7,000, and 10,000 in sales
E(2,000) = 0.15(Max(0,2000 - 5000)) + 300
E(2,000) = 0.15(Max(0,-3000)) + 300
E(2,000) = 0.15(0) + 300
[B]E(2,000) = 300
[/B]
E(7,000) = 0.15(Max(0,7000 - 5000)) + 300
E(7,000) = 0.15(Max(0,2000)) + 300
E(7,000) = 0.15(2,000) + 300
E(7,000) = 300 + 300
[B][B]E(7,000) = 600[/B][/B]
E(10,000) = 0.15(Max(0,10000 - 5000)) + 300
E(10,000) = 0.15(Max(0,5000)) + 300
E(10,000) = 0.15(5,000) + 300
E(10,000) = 750+ 300
[B][B]E(10,000) = 1,050[/B][/B]

A salesperson works 40 hours per week at a job where he has two options for being paid. Option A is

A salesperson works 40 hours per week at a job where he has two options for being paid. Option A is an hourly wage of $24. Option B is a commission rate of 4% on weekly sales. How much does he need to sell this week to earn the same amount with the two options?
Option A payment function:
24h
With a 40 hour week, we have:
24 * 40 = 960
Option B payment function with sales amount (s):
0.04s
We want to know the amount of sales (s) where Option A at 40 hours = Option B. So we set both equal to each other:
0.04s = 960
To solve this equation for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.04s%3D960&pl=Solve']type it in our math engine[/URL] and we get:
s = [B]24,000[/B]

A school theater group is selling candy to raise funds in order to put on their next performance. Th

A school theater group is selling candy to raise funds in order to put on their next performance. The candy cost the group $0.20 per piece. Plus, there was a $9 shipping and handling fee. The group is going to sell the candy for $0.50 per piece. How many pieces of candy must the group sell in order to break even?
[U]Set up the cost function C(p) where p is the number of pieces of candy.[/U]
C(p) = Cost per piece * p + shipping and handling fee
C(p) = 0.2p + 9
[U]Set up the Revenue function R(p) where p is the number of pieces of candy.[/U]
R(p) = Sale price * p
R(p) = 0.5p
Break-even means zero profit or loss, so we set the Cost Function equal to the Revenue Function
0.2p + 9 = 0.5p
To solve this equation for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.2p%2B9%3D0.5p&pl=Solve']type it in our math engine[/URL] and we get:
p = [B]30[/B]

A shipping service charges $0.43 for the first ounce and $0.29 for each additional ounce of package

A shipping service charges $0.43 for the first ounce and $0.29 for each additional ounce of package weight. Write an equation to represent the price P of shipping a package that weighs x ounces, for any whole number of ounces greater than or equal to 1.
Set up the price function P(x)
[B]P(x) = 0.43 + 0.29(x - 1)[/B]

A soccer team is buying T-shirts to sell as a fundraiser. The team pays a flat fee of $35 for a logo

A soccer team is buying T-shirts to sell as a fundraiser. The team pays a flat fee of $35 for a logo design plus $7.00 per T-shirt.
Set up the cost function C(t) where t is the number of t-shirts:
C(t) = Cost per t-shirt * number of t-shirts + Flat Fee
[B]C(t) = 7t + 35[/B]

A spherical water tank holds 11,500ft^3 of water. What is the diameter?

A spherical water tank holds 11,500ft^3 of water. What is the diameter?
The tank holding amount is volume. And the volume of a sphere is:
V = (4pir^3)/3
We know that radius is 1/2 of diameter:
r =d/2
So we rewrite our volume function:
V = 4/3(pi(d/2)^3)
We're given V = 11,500 so we have:
4/3(pi(d/2)^3) = 11500
Multiply each side by 3/4
4/3(3/4)(pi(d/2)^3) = 11,500*3/4
Simplify:
pi(d/2)^3 = 8625
Since pi = 3.1415926359, we divide each side by pi:
(d/2)^3 = 8625/3.1415926359
(d/2)^3 = 2745.42
Take the cube root of each side:
d/2 = 14.0224
Multiply through by 2:
[B]d = 28.005[/B]

A student and the marine biologist are together at t = 0. The student ascends more slowly than the m

A student and the marine biologist are together at t = 0. The student ascends more slowly than the marine biologist. Write an equation of a function that could represent the student's ascent. Please keep in mind the slope for the marine biologist is 12.
Slope means rise over run.
In this case, rise is the ascent distance and run is the time.
12 = 12/1, so for each second of time, the marine biologist ascends 12 units of distance
If the student ascends slower, than the total distance gets reduced by an unknown factor, let's call it c. So we have the student's ascent function as:
[B]y(t) = 12t - c[/B]

a student has $50 in saving and earns $40 per week. How long would it take them to save $450

a student has $50 in saving and earns $40 per week. How long would it take them to save $450
Set up the savings function S(w), where w is the number of weeks. The balance, S(w) is:
S(w) = Savings Per week * w + Initial Savings
S(w) = 40w + 50
The problems asks for how many weeks for S(w) = 450. So we have;
40w + 50 = 450
To solve for w, we[URL='https://www.mathcelebrity.com/1unk.php?num=40w%2B50%3D450&pl=Solve'] type this equation in our search engine[/URL] and we get:
w = [B]10[/B]

A super deadly strain of bacteria is causing the zombie population to double every day. Currently, t

A super deadly strain of bacteria is causing the zombie population to double every day. Currently, there are 25 zombies. After how many days will there be over 25,000 zombies?
We set up our exponential function where n is the number of days after today:
Z(n) = 25 * 2^n
We want to know n where Z(n) = 25,000.
25 * 2^n = 25,000
Divide each side of the equation by 25, to isolate 2^n:
25 * 2^n / 25 = 25,000 / 25
The 25's cancel on the left side, so we have:
2^n = 1,000
Take the natural log of each side to isolate n:
Ln(2^n) = Ln(1000)
There exists a logarithmic identity which states: Ln(a^n) = n * Ln(a). In this case, a = 2, so we have:
n * Ln(2) = Ln(1,000)
0.69315n = 6.9077
[URL='https://www.mathcelebrity.com/1unk.php?num=0.69315n%3D6.9077&pl=Solve']Type this equation into our search engine[/URL], we get:
[B]n = 9.9657 days ~ 10 days[/B]

A taxi cab in Chicago charges $3 per mile and $1 for every person. If the taxi cab ride for two peop

A taxi cab in Chicago charges $3 per mile and $1 for every person. If the taxi cab ride for two people costs $20. How far did the taxi cab travel.
Set up a cost function C(m) where m is the number of miles driven:
C(m) = cost per mile * m + per person fee
[U]Calculate per person fee:[/U]
per person fee = $1 per person * 2 people
per person fee = $2
[U]With a cost per mile of $3 and per person fee of $2, we have:[/U]
C(m) = cost per mile * m + per person fee
C(m) = 3m + 2
The problem asks for m when C(m) = 20, so we set 3m + 2 equal to 20:
3m + 2 = 20
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=3m%2B2%3D20&pl=Solve']plug it in our search engine[/URL] and we get:
m = [B]6[/B]

A taxi cab in nyc charges a pick up fee of $5.00 . The customer must also pay $2.59 for each mile th

A taxi cab in nyc charges a pick up fee of $5.00 . The customer must also pay $2.59 for each mile that the taxi must drive to reach their destination. Write an equation
Set up a cost function C(m) where m is the number of miles:
C(m) = Mileage Charge * m + pick up fee
[B]C(m) = 2.59m + 5[/B]

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most $10 to

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most $10 to spend on the cab ride, how far could she travel?
Set up a cost function C(m), where m is the number of miles:
C(m) = Cost per mile * m + flat rate
C(m) = 0.65m + 1.75
The problem asks for m when C(m) = 10
0.65m + 1.75 = 10
[URL='https://www.mathcelebrity.com/1unk.php?num=0.65m%2B1.75%3D10&pl=Solve']Typing this equation into the search engine[/URL], we get:
m = [B]12.692 miles[/B]

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most 10$ to

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most 10$ to spend on the cab ride, how far could she travel
Set up a cost function C(m), where m is the number of miles Erica can travel. We have:
C(m) = 0.65m + 1.75
If C(m) = 10, we have:
0.65m + 1.75 = 10
[URL='https://www.mathcelebrity.com/1unk.php?num=0.65m%2B1.75%3D10&pl=Solve']Typing this equation into our search engine[/URL], we get:
m = 12.69 miles
If the problem asks for complete miles, we round down to 12 miles.

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most 10$ to

A taxi charges a flat rate of $1.75, plus an additional $0.65 per mile. If Erica has at most 10$ to spend on the cab ride, how far could she travel?
Set up the cost function C(m) where m is the number of miles:
C(m) = 0.65m + 1.75
If Erica has $10, then C(m) = 10, so we have:
0.65m + 1.75 = 10
[URL='https://www.mathcelebrity.com/1unk.php?num=0.65m%2B1.75%3D10&pl=Solve']Typing this equation into the search engine[/URL], we get
m = 12.69
if the answer asks for whole number, then we round down to m = 12

A text message plan costs $7 per month plus $0.28 per text. Find the monthly cost for x text message

A text message plan costs $7 per month plus $0.28 per text. Find the monthly cost for x text messages.
We set up the cost function C(x) where x is the number of text messages per month:
C(x) = Cost per text * x + Monthly cost
Plugging in our given numbers, we get:
[B]C(x) = 0.28x + 7[/B]

A towns population is currently 500. If the population doubles every 30 years, what will the populat

A towns population is currently 500. If the population doubles every 30 years, what will the population be 120 years from now?
Find the number of doubling times:
120 years / 30 years per doubling = 4 doubling times
Set up our growth function P(n) where n is the number of doubling times:
P(n) = 500 * 2^n
Since we have 4 doubling times, we want P(4):
P(4) = 500 * 2^4
P(4) = 500 * 16
P(4) = [B]8,000[/B]

A toy company makes "Teddy Bears". The company spends $1500 for factory expenses plus $8 per bear. T

A toy company makes "Teddy Bears". The company spends $1500 for factory expenses plus $8 per bear. The company sells each bear for $12.00 each. How many bears must this company sell in order to break even?
[U]Set up the cost function C(b) where b is the number of bears:[/U]
C(b) = Cost per bear * b + factory expenses
C(b) = 8b + 1500
[U]Set up the revenue function R(b) where b is the number of bears:[/U]
R(b) = Sale Price per bear * b
R(b) = 12b
[U]Break-even is where cost equals revenue, so we set C(b) equal to R(b) and solve for b:[/U]
C(b) = R(b)
8b + 1500 = 12b
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=8b%2B1500%3D12b&pl=Solve']type this equation into our search engine[/URL] and we get:
b = [B]375[/B]

A video store charges a monthly membership fee of $7.50, but the charge to rent each movie is only $

A video store charges a monthly membership fee of $7.50, but the charge to rent each movie is only $1.00 per movie. Another store has no membership fee, but it costs $2.50 to rent each movie. How many movies need to be rented each month for the total fees to be the same from either company?
Set up a cost function C(m) where m is the number of movies you rent:
C(m) = Rental cost per movie * m + Membership Fee
[U]Video Store 1 cost function[/U]
C(m) = 1m + 7.5
Video Store 2 cost function:
C(m) = 2.50m
We want to know when the costs are the same. So we set each C(m) equal to each other:
m + 7.5 = 2.50m
To solve this equation for m, [URL='https://www.mathcelebrity.com/1unk.php?num=m%2B7.5%3D2.50m&pl=Solve']we type it in our search engine[/URL] and we get:
m = [B]5[/B]

a well driller charges $9.00 per foot for the first 10 feet, 9.10 per foot for the next 10 feet, $9.

a well driller charges $9.00 per foot for the first 10 feet, 9.10 per foot for the next 10 feet, $9.20 per foot for the next 10 feet, and so on, at a price increase of $0.10 per foot for succeeding intervals of 10 feet. How much does it cost to drill a well to a depth of 150 feet?
Set up the cost function C(f) where f is the number of feet:
Cost = 9(10) + 9.1(10) + 9.2(10) + 9.3(10) + 9.4(10) + 9.5(10) + 9.6(10) + 9.7(10) + 9.8(10) + 9.9(10) + 10(10) + 10.1(10) + 10.2(10) + 10.3(10) + 10.4(10)
Cost = [B]1,455[/B]

a writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel in

a writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel in x number of pages, determine a function model that will represent the accumulated writing hours to finish his novel
if 3 pages = 5 hours, then we divide each side by 3 to get:
1 page = 5/3 hours per page
So x pages takes:
5x/3 hours
Our function for number of pages x is:
[B]f(x) = 5x/3[/B]

A yoga member ship costs $16 and additional $7 per class. Write a linear equation modeling the cost

A yoga member ship costs $16 and additional $7 per class. Write a linear equation modeling the cost of a yoga membership?
Set up the cost function M(c) for classes (c)
[B]M(c) = 16 + 7c[/B]

Adam, Bethany, and Carla own a painting company. To paint a particular home, Adam estimates it woul

Adam, Bethany, and Carla own a painting company. To paint a particular home, Adam estimates it would take him 4 days. Bethany estimates 5.5 days. Carla estimates 6 days. How long would it take them to work together to paint the house.
Our combined work function for time (t) using a = Adam's time, b = Bethany's time, and c = Carla's time is:
1/a + 1/b + 1/c = 1/t
Plugging in a, b, and c, we get:
1/4 + 1/5.5 + 1/6 = 1/t
0.25 + 0.181818 + 0.1667 = 1/t
1/t = 0.59848
t = [B]1.67089 days[/B]

Alorah joins a fitness center. She pays for a year plus a joining fee of $35. If the cost for the en

Alorah joins a fitness center. She pays for a year plus a joining fee of $35. If the cost for the entire year is $299, how much will she pay each month?
We set up the cost function C(m) where m is the number of months of membership:
C(m) = cost per month * m + joining fee
Plugging in our numbers from the problem with 12 months in a year, we get:
12c + 35 = 299
To solve this equation for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=12c%2B35%3D299&pl=Solve']type it in our search engine [/URL]and we get:
c = [B]22[/B]

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for ea

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Let the number of tv's be t. Set up the salary function S(t):
S(t) = Commision * tv's sold + Salary
Company A:
S(t) = 100t + 17,000
Company B:
S(t) = 20t + 29,000
The problem asks for how many tv's it takes to make both company salaries equal. So we set the S(t) functions equal to each other:
100t + 17000 = 20t + 29000
[URL='https://www.mathcelebrity.com/1unk.php?num=100t%2B17000%3D20t%2B29000&pl=Solve']Type this equation into our search engine[/URL] and we get:
t = [B]150[/B]

Amy and ryan operate a car dealing and repair service. For a car detailing (full wash outside and in

Amy and ryan operate a car dealing and repair service. For a car detailing (full wash outside and inside. Amy charges 40$ and Ryan charges 50$ . In addition they charge a hourly rate. Amy charges $35/h and ryan charges $30/h. How many hours does amy and ryan have to work to make the same amount of money?
Set up the cost functions C(h) where h is the number of hours.
[U]Amy:[/U]
C(h) = 35h + 40
[U]Ryan:[/U]
C(h) = 30h + 50
To make the same amount of money, we set both C(h) functions equal to each other:
35h + 40 = 30h + 50
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=35h%2B40%3D30h%2B50&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]2[/B]

An airplane is flying at 38,800 feet above sea level. The airplane starts to descend at a rate of 18

An airplane is flying at 38,800 feet above sea level. The airplane starts to descend at a rate of 1800 feet per minute. Let m be the number of minutes. Which of the following expressions describe the height of the airplane after any given number of minutes?
Let m be the number of minutes. Since a descent equals a [U]drop[/U] in altitude, we subtract this in our Altitude function A(m):
[B]A(m) = 38,800 - 1800m[/B]

An auto repair bill is $126 for parts and $35 for each hour of labor. If h is the number of hours of

An auto repair bill is $126 for parts and $35 for each hour of labor. If h is the number of hours of labor, express the amount of the repair bill in terms of number of hours of labor.
Set up cost function, where h is the number of hours of labor:
[B]C(h) = 35h + 136[/B]

An auto repair bill was $563. This includes $188 for parts and $75 for each hour of labor. Find the

An auto repair bill was $563. This includes $188 for parts and $75 for each hour of labor. Find the number of hours of labor
Let the number of hours of labor be h. We have the cost function C(h):
C(h) = Hourly Labor Rate * h + parts
Given 188 for parts, 75 for hourly labor rate, and 563 for C(h), we have:
75h + 188 = 563
To solve this equation for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=75h%2B188%3D563&pl=Solve']type it in our search engine[/URL] and we get:
h = [B]5[/B]

An interior designer charges $100 to visit a site, plus $55 to design each room. Identify a function

An interior designer charges $100 to visit a site, plus $55 to design each room. Identify a function that represents the total amount he charges for designing a certain number of rooms. What is the value of the function for an input of 6, and what does it represent?
[U]Set up the cost function C(r) where r is the number of room to design:[/U]
C(r) = Cost per room * r + Site Visit Fee
C(r) = 55r + 100
[U]Now, the problem asks for an input of 6, which is [I]the number of rooms[/I]. So we want C(6) which is the [I]cost to design 6 rooms[/I]:[/U]
C(6) = 55(6) + 100
C(6) = 330 + 100
C(6) = [B]430[/B]

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community col

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community college charges a $35 registration fee plus $375 per course, what is the greatest number of courses for which Angelica can register?
We set up the Tuition function T(c), where c is the number of courses:
T(c) = Cost per course * c + Registration Fee
T(c) = 35c + 375
The problem asks for the number of courses (c) where her tuition [I]cannot exceed[/I] $1000. The phrase [I]cannot exceed[/I] means less than or equal to, or no more than. So we setup the inequality for T(c) <= 1000 below:
35c + 375 <= 1000
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=35c%2B375%3C%3D1000&pl=Solve']type it in our search engine and we get[/URL]:
c <= 17.85
Since we cannot have fractional courses, we round down and get:
c[B] <= 17[/B]

Ann took a taxi home from the airport. The taxi fare was $2.10 per mile, and she gave the driver a t

Ann took a taxi home from the airport. The taxi fare was $2.10 per mile, and she gave the driver a tip of $5 Ann paid a total of $49.10.
Set up the cost function C(m) where m is the number of miles:
C(m) = Mileage Rate x m + Tip
2.10m + 5 = 49.10
[URL='https://www.mathcelebrity.com/1unk.php?num=2.10m%2B5%3D49.10&pl=Solve']Type 2.10m + 5 = 49.10 into the search engine[/URL], and we get [B]m = 21[/B].

At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonme

At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same?
Set up the cost functions where x is the number of aerobics classes:
[LIST]
[*]Members: C(x) = 10 + 3x
[*]Non-members: C(x) = 5x
[/LIST]
Set them equal to each other
10 + 3x = 5x
Subtract 3x from both sides:
2x = 10
Divide each side by 2
[B]x = 5 classes[/B]

Belle bought 30 pencils for $1560. She made a profit of $180. How much profit did she make on each p

Belle bought 30 pencils for $1560. She made a profit of $180. How much profit did she make on each pencil
The cost per pencil is:
1560/30 = 52
Build revenue function:
Revenue = Number of Pencils * Sales Price (s)
Revenue = 30s
The profit equation is:
Profit = Revenue - Cost
Given profit is 180 and cost is 1560, we have:
30s - 1560 = 180
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=30s-1560%3D180&pl=Solve']type this equation into our search engine[/URL] and we get:
s = 58
This is sales for total profit. The question asks profit per pencil.
Profit per pencil = Revenue per pencil - Cost per pencil
Profit per pencil = 58 - 52
Profit per pencil = [B]6[/B]

Bike rental shop A charges $20 per kilometre travelled with no additional fee. Bike rental shop B ch

Bike rental shop A charges $20 per kilometre travelled with no additional fee. Bike rental shop B charges only $8 per kilometre travelled, but has a starting charge of $35. If Bob plans to travel 7km by bike, which rental shop should he choose for a better price
[U]Shop A Cost function C(k) where k is the number of kilometers used[/U]
C(k) = Cost per kilometer * k + Starting Charge
C(k) = 20k
With k = 7, we have:
C(7) = 20 * 7
C(7) = 140
[U]Shop B Cost function C(k) where k is the number of kilometers used[/U]
C(k) = Cost per kilometer * k + Starting Charge
C(k) = 8k + 35
With k = 7, we have:
C(7) = 8 * 7 + 35
C(7) = 56 + 35
C(7) = 91
Bog should choose [B]Shop B[/B] since they have the better price for 7km

Bills car rental charges a base fee of 50$ and then $0.20 per mile

Bills car rental charges a base fee of 50$ and then $0.20 per mile.
Set up the cost function C(m) where m is the number of miles driven:
[B]C(m) = 50 + 0.20m[/B]

Binomial Distribution

Calculates the probability of 3 separate events that follow a binomial distribution. It calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness and kurtosis.

Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor

Calculates moment number t using the moment generating function

Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor

Calculates moment number t using the moment generating function

Break Even

Given a fixed cost, variable cost, and revenue function or value, this calculates the break-even point

Can a coefficient of determination be negative? Why or why not?

Can a coefficient of determination be negative? Why or why not?
[B]Yes, reasons below[/B]
[LIST]
[*] predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data
[*] where linear regression is conducted without including an intercept
[*] Yes, negative values of R2 may occur when fitting non-linear functions to data
[/LIST]

Carly has already written 35 of a novel. She plans to write 12 additional pages per month until she

Carly has already written 35 of a novel. She plans to write 12 additional pages per month until she is finished. Write and solve a linear equation to find the total number of pages written at 5 months.
Let m be the number of months. We have the pages written function P(m) as:
P(m) = 12m + 35
The problem asks for P(5):
P(5) = 12(5) + 35
P(5) = 60 + 35
P(5) = [B]95[/B]

Cofunction Calculator

Calculates the cofunction of the 6 trig functions:
* sin

* cos

* tan

* csc

* sec

* cot

* cos

* tan

* csc

* sec

* cot

Compound Interest and Annuity Table

Given an interest rate (i), number of periods to display (n), and number of digits to round (r), this calculator produces a compound interest table. It shows the values for the following 4 compound interest annuity functions from time 1 to (n) rounded to (r) digits:

v^{n}

d

(1 + i)^{n}

a_{n|}

s_{n|}

ä_{n|i}

s_{n|i}

Force of Interest δ^{n}

v

d

(1 + i)

a

s

ä

s

Force of Interest δ

Congratulations!! You are hired at Roof and Vinyl Housing Systems. Your starting salary is $45,600 f

Congratulations!! You are hired at Roof and Vinyl Housing Systems. Your starting salary is $45,600 for the year. Each year you stay employed with them your salary will increase by 3.5%. Determine what your salary would be if you worked for the company for 12 years.
Set up a function S(y) where y is the number of years after you start at the Roof and Vinyl place.
S(y) = 45600 * (1.035)^y <-- Since 3.5% = 0.035
The question asks for S(12):
S(12) = 45600 * (1.035)^12
S(12) = 45600 * 1.51106865735
S(12) = [B]68,904.73[/B]

Cost Function

Creates a cost function from cost word problems

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What w

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What was the greatest number of games Craig could have played?
Set up the cost function C(g) where g is the number of games Craig plays:
C(g) = Game fee * number of games (g) + shoe rental fee
C(g) = 4g + 5.25
The problem asks for the maximum number of games Craig can play for $25. So we want an inequality of [I]less than or equal to[/I].
4g + 5.25 <= 25
[URL='https://www.mathcelebrity.com/1unk.php?num=4g%2B5.25%3C%3D25&pl=Solve']Type this inequality into our search engine[/URL], and we get:
g <= 4.9375
We want exact games, so we round this down to [B]4 games[/B].

Critical Z-values

Given a probability from a normal distribution, this will generate the z-score critical value. Uses the NORMSINV Excel function.

DeAndre is a spelunker (someone who explores caves). One day DeAndre is exploring a cave that has a

DeAndre is a spelunker (someone who explores caves). One day DeAndre is exploring a cave that has a series of ladders going down into the depths. Every ladder is exactly 10 feet tall, and there is no other way to descend or ascend (the other paths in the cave are flat). DeAndre starts at 186 feet in altitude, and reaches a maximum depth of 86 feet in altitude.Write an equation for DeAndre's altitude, using x to represent the number of ladders DeAndre used (hint: a ladder takes DeAndre down in altitude, so the coefficient should be negative).
Set up a function A(x) for altitude, where x is the number of ladders used. Each ladder takes DeAndre down 10 feet, so this would be -10x. And DeAndre starts at 186 feet, so we'd have:
[B]A(x) = 186 - 10x[/B]

Demand curve of a software development firm digisign is qd= -16p^2 -4p + 250 .Carry out analysis

Demand curve of a software development firm digisign is
qd= -16p^2 -4p + 250 .Carry out analysis of demand function and find out elasticity of demand Ed At p=25.
Draw pertinent conclusions .
qd(25) = -16*(25^2) - 4(25) + 250
qd(25) = -10000 - 100 + 250
qd(25) = -9850
So p = 25 and q = -9850
Ed = dp/dq
Ed = -13p -4
Elasticity of Demand is found at:
Ed = p/q * dp/dq
Ed = 25/-9850 * (-13(25) - 4)
Ed = 25/-9850 * -329
Ed = [B]0.835[/B]

Diana earns $8.50 working as a lifeguard. Write an equation to find Dianas money earned m for any nu

Diana earns $8.50 working as a lifeguard. Write an equation to find Dianas money earned m for any numbers of hours h
Set up the revenue function:
[B]R = 8.5h[/B]

Diego is jogging at a rate of 5mi/h. A function relates how far Deigo jogs to his rate of speed.

Let d be distance and h be hours in time. Set up our function.
[LIST]
[*]f(h) = d
[*][B]f(h) = 5h[/B]
[/LIST]
Read this out, it says, for every hour Diego jogs, multiply that by 5 to get the distance he jogs.

Dotty McGinnis starts up a small business manufacturing bobble-head figures of famous soccer players

Dotty McGinnis starts up a small business manufacturing bobble-head figures of famous soccer players. Her initial cost is $3300. Each figure costs $4.50 to make. a. Write a cost function, C(x), where x represents the number of figures manufactured.
Cost function is the fixed cost plus units * variable cost.
[B]C(x) = 3300 + 4.50x[/B]

Dr. Carlson is contemplating the impact of an antibiotic on a particular patient. The patient will t

Dr. Carlson is contemplating the impact of an antibiotic on a particular patient. The patient will take 229 milligrams, and every hour his body will break down 20% of it. How much will be left after 9 hours?
Set up the antibiotic remaining function A(h) where h is the number of hours after the patient takes the antibiotic.
If the body breaks down 20%, then the remaining is 100% - 20% = 80%
80% as a decimal is 0.8, so we have:
A(h) = 229 * (0.8)^h
The problems asks for A(9):
A(9) = 229 * (0.8)^9
A(9) = 229 * 0.134217728
A(9) = [B]30.74 milligrams[/B]

Dr. Hoffman is contemplating the impact of an antibiotic on a particular patient. The patient will t

Dr. Hoffman is contemplating the impact of an antibiotic on a particular patient. The patient will take 590 milligrams, and every hour his body will break down 30% of it. How much will be left after 8 hours? If necessary, round your answer to the nearest tenth.
Set up a function A(h), where h is the number of hours since the patient took the antibiotic.
If the body breaks down 30%, it keeps 70%, or 0.7.
A(h) = 590(0.70)^h
The problem asks for A(8):
A(8) = 590(0.70)^8
A(8) =590 * 0.05764801
A(8) = 34.012 hours
Rounded to the nearest tenth, it's [B]34.0 hours[/B].

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers business cards for $0.15 each with a setup charge of $10. What numbers of business cards cost the same from either company
Declare variables:
[LIST]
[*]Let b be the number of business cards.
[/LIST]
[U]Set up the cost function C(b) for Dunder Mifflin:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.1b + 15
[U]Set up the cost function C(b) for Werham Hogg:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.15b + 10
The phrase [I]cost the same[/I] means we set both C(b)'s equal to each other and solve for b:
0.1b + 15 = 0.15b + 10
Solve for [I]b[/I] in the equation 0.1b + 15 = 0.15b + 10
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.1b and 0.15b. To do that, we subtract 0.15b from both sides
0.1b + 15 - 0.15b = 0.15b + 10 - 0.15b
[SIZE=5][B]Step 2: Cancel 0.15b on the right side:[/B][/SIZE]
-0.05b + 15 = 10
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 15 and 10. To do that, we subtract 15 from both sides
-0.05b + 15 - 15 = 10 - 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
-0.05b = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -0.05[/B][/SIZE]
-0.05b/-0.05 = -5/-0.05
b = [B]100[/B]

During a performance, a juggler tosses one ball straight upward while continuing to juggle three oth

During a performance, a juggler tosses one ball straight upward while continuing to juggle three others. The height f(t), in feet, of the ball is given by the polynomial function f(t) = ?16t^2 + 26t + 3, where t is the time in seconds since the ball was thrown. Find the height of the ball 1 second after it is tossed upward.
We want f(1):
f(1) = ?16(1)^2 + 26(1) + 3
f(1) = -16(1) + 26 + 3
f(1) = -16 + 26 + 3
f(1) = [B]13[/B]

evelyn needs atleast $112 to buy a new dress. She has already saved $40 . She earns $9 an hour babys

evelyn needs atleast $112 to buy a new dress. She has already saved $40 . She earns $9 an hour babysitting. How many hours will she need to babysit to buy the dress?
Let the number of hours be h.
We have the earnings function E(h) below
E(h) = hourly rate * h + current savings
E(h) = 9h + 40
We're told E(h) = 112, so we have:
9h + 40 = 112
[URL='https://www.mathcelebrity.com/1unk.php?num=9h%2B40%3D112&pl=Solve']Typing this equation in our math engine[/URL] and we get:
h = [B]8[/B]

Exponential Distribution

Calculates the Probability Density Function (PDF) and Cumulative Density Function (CDF) of the exponential distribution as well as the mean, variance, standard deviation, and entropy.

f(x) = 3x - 1; g(x) = 15 - 3*f(x) The functions f and g are defined above. What is the value of g(2)

f(x) = 3x - 1; g(x) = 15 - 3*f(x) The functions f and g are defined above. What is the value of g(2)?
g(2) = 15 - 3 * f(2)
f(2) = 3(2) - 1
f(2) = 6 - 1
f(2) = 5
Therefore, with f(2) = 5, g(2) is:
g(2) = 15 - 3 * f(2)
g(2) = 15 - 3 * 5
g(2) = 15 - 15
g(2) = [B]0[/B]

Farmer Yumi has too many plants in her garden. If she starts out with 150 plants and is losing them

Farmer Yumi has too many plants in her garden. If she starts out with 150 plants and is losing them at a rate of 4% each day, how long will it take for her to have 20 plants left? Round UP to the nearest day.
We set up the function P(d) where d is the number of days sine she started losing plants:
P(d) = Initial plants * (1 - Loss percent / 100)^d
Plugging in our numbers, we get:
20 = 150 * (1 - 4/100)^d
20 = 150 * (1 - 0.04)^d
Read left to right so it's easier to read:
150 * 0.96^d = 20
Divide each side by 150, and we get:
0.96^d = 0.13333333333
To solve this logarithmic equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=0.96%5Ed%3D0.13333333333&pl=Calculate']type it in our search engine[/URL] and we get:
d = 49.35
The problem tells us to round up, so we round up to [B]50 days[/B]

Find a linear function f, given f(16)=-2 and f(-12)=-9. Then find f(0)

Find a linear function f, given f(16)=-2 and f(-12)=-9. Then find f(0).
We've got 2 points:
(16, -2) and (-12, -9)
Calculate the slope (m) of this line using:
m = (y2 - y1)/(x2 - x1)
m = (-9 - -2)/(-12 - 16)
m = -7/-28
m = 1/4
The line equation is denoted as:
y = mx + b
Let's use the first point (x, y) = (16, -2)
-2 = 1/4(16) + b
-2 = 4 + b
Subtract 4 from each side, and we get:
b = -6
So our equation of the line is:
y = 1/4x - 6
The questions asks for f(0):
y = 1/4(0) - 6
y = 0 - 6
[B]y = -6[/B]

Fixed cost 500 marginal cost 8 item sells for 30

fixed cost 500 marginal cost 8 item sells for 30.
Set up Cost Function C(x) where x is the number of items sold:
C(x) = Marginal Cost * x + Fixed Cost
C(x) = 8x + 500
Set up Revenue Function R(x) where x is the number of items sold:
R(x) = Revenue per item * items sold
R(x) = 30x
Set up break even function (Cost Equals Revenue)
C(x) = R(x)
8x + 500 = 30x
Subtract 8x from each side:
22x = 500
Divide each side by 22:
x = 22.727272 ~ 23 units for breakeven

Flight is $295 and car rental is $39 a day, if a competition charges $320 and $33 a day car rental,

Flight is $295 and car rental is $39 a day, if a competition charges $320 and $33 a day car rental, which is cheaper?
Set up cost function where d is the number of days:
[LIST]
[*]Control business: C(d) = 39d + 295
[*]Competitor business: C(d) = 33d + 320
[/LIST]
Set the [URL='http://www.mathcelebrity.com/1unk.php?num=39d%2B295%3D33d%2B320&pl=Solve']cost functions equal to each other[/URL]:
We get d = 4.1667.
The next integer day up is 5. Now plug in d = 1, 2, 3, 4. For the first 4 days, the control business is cheaper. However, starting at day 5, the competitor business is now cheaper forever.

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minu

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minute of use. The least she has been charged in a month is $86.04 . What are the possible numbers of minutes she has used her phone in a month? Use m for the number of minutes, and solve your inequality for m .
Maya's cost function is C(m), where m is the number of minutes used.
C(m) = 0.04m + 27
We are given C(m) = $86.04. We want her cost function [I]less than or equal[/I] to this.
0.04m + 27 <= 86.04
[URL='https://www.mathcelebrity.com/1unk.php?num=0.04m%2B27%3C%3D86.04&pl=Solve']Type this inequality into our search engine[/URL], and we get [B]m <= 1476[/B].

for the function, h(x) = bx - 22, b is a constant and h(-5) = -7. Find the value of h(5)

for the function, h(x) = bx - 22, b is a constant and h(-5) = -7. Find the value of h(5)
h(-5) = -5b - 22
Since we're given h(-5) = -7, we have:
-5b - 22 = -7
[URL='https://www.mathcelebrity.com/1unk.php?num=-5b-22%3D-7&pl=Solve']Typing this equation into our search engine[/URL], we get:
b = -3
So our h(x) equation is now:
h(x) = -3x - 22
The problem asks for h(5):
h(5) = -3(5) - 22
h(5) = 15 - 22
h(5) = [B]-37[/B]

Frank is a plumber who charges a $35 service charge and $15 per hour for his plumbing services. Find

Frank is a plumber who charges a $35 service charge and $15 per hour for his plumbing services. Find a linear function that expresses the total cost C for plumbing services for h hours.
Cost functions include a flat rate and a variable rate. The flat rate is $35 and the variable rate per hour is 15. The cost function C(h) where h is the number of hours Frank works is:
[B]C(h) = 15h + 35[/B]

Function

Takes various functions (exponential, logarithmic, signum (sign), polynomial, linear with constant of proportionality, constant, absolute value), and classifies them, builds ordered pairs, and finds the y-intercept and x-intercept and domain and range if they exist.

Function Test

Checks to see if a set of ordered pairs represents a function

Functions-Derivatives-Integrals

Given a polynomial expression, this calculator evaluates the following items:

1) Functions ƒ(x). Your expression will also be evaluated at a point, i.e., ƒ(1)

2) 1^{st} Derivative ƒ'(x) The derivative of your expression will also be evaluated at a point, i.e., ƒ'(1)

3) 2^{nd} Derivative ƒ''(x) The second derivative of your expression will be also evaluated at a point, i.e., ƒ''(1)

4) Integrals ∫ƒ(x) The integral of your expression will also be evaluated on an interval, i.e., [0,1]

5) Using Simpsons Rule, the calculator will estimate the value of ≈ ∫ƒ(x) over an interval, i.e., [0,1]

1) Functions ƒ(x). Your expression will also be evaluated at a point, i.e., ƒ(1)

2) 1

3) 2

4) Integrals ∫ƒ(x) The integral of your expression will also be evaluated on an interval, i.e., [0,1]

5) Using Simpsons Rule, the calculator will estimate the value of ≈ ∫ƒ(x) over an interval, i.e., [0,1]

Gabe rents a piano for $49 per month. He earns $15 per hour giving piano lessons to students. How ma

Gabe rents a piano for $49 per month. He earns $15 per hour giving piano lessons to students. How many hours of lessons per month must he give to earn a profit of $326?
Build a profit function P(h) where h is the number of hours:
P(h) = Hourly Rate * Number of Hours (h) - Cost of Piano
P(h) = 15h - 49
The problem asks for the number of hours where P(h) = $326
15h - 49 = 326
We take this equation and [URL='https://www.mathcelebrity.com/1unk.php?num=15h-49%3D326&pl=Solve']type it in our search engine[/URL] to solve for h:
h = [B]25[/B]

Geometric Distribution

Using a geometric distribution, it calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

Calculates moment number t using the moment generating function

Given the function f(x)=3x?9, what is the value of x when f(x)=9

Given the function f(x)=3x?9, what is the value of x when f(x)=9
Plug in our numbers and we get:
3x - 9 = 9
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=3x-9%3D9&pl=Solve']type it in our search engine[/URL] and we get:
x = [B]6[/B]

Graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute.Max is hiking a

Graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute.Max is hiking at an altitude of 12,500 feet and is ascending 20 feet each minute. How many minutes will it take until they're at the same altitude?
Set up the Altitude function A(m) where m is the number of minutes that went by since now.
Set up Graham's altitude function A(m):
A(m) = 14040 - 50m <-- we subtract for descending
Set up Max's altitude function A(m):
A(m) = 12500 + 20m <-- we add for ascending
Set the altitudes equal to each other to solve for m:
14040 - 50m = 12500 + 20m
[URL='https://www.mathcelebrity.com/1unk.php?num=14040-50m%3D12500%2B20m&pl=Solve']We type this equation into our search engine to solve for m[/URL] and we get:
m = [B]22[/B]

Gym A: $75 joining fee and $35 monthly charge. Gym B: No joining fee and $60 monthly charge. (Think

Gym A: $75 joining fee and $35 monthly charge. Gym B: No joining fee and $60 monthly charge. (Think of the monthly charges paid at the end of the month.) Enter the number of months it will take for the total cost for both gyms to be equal.
Gym A cost function C(m) where m is the number of months:
C(m) = Monthly charge * months + Joining Fee
C(m) = 35m + 75
Gym B cost function C(m) where m is the number of months:
C(m) = Monthly charge * months + Joining Fee
C(m) = 60m
Set them equal to each other:
35m + 75 = 60m
To solve for m, [URL='https://www.mathcelebrity.com/1unk.php?num=35m%2B75%3D60m&pl=Solve']we type this equation into our search engine[/URL] and get:
m = [B]3[/B]

Hyperbolic Function

Calculates hyperbolic function values:
sinh, cosh, tanh, csch, sech, coth

Hyperbolic Inverse

Calculates hyperbolic function values:
arcsinh, arccosh, arctanh, arccsch, arcsech, arccoth

I have $36 dollars and it goes up by 3 every day how much money would I have after 500 days

I have $36 dollars and it goes up by 3 every day how much money would I have after 500 days
We have a balance function B(d) where d is the number of days passed since we first had $36:
B(d) = 3d + 36
The problem asks for B(500):
B(500) = 3(500) + 36
B(500) = 1500 + 36
B(500) = [B]1536[/B]

If 200 bacteria triple every 1/2 hour, how much bacteria in 3 hours

If 200 bacteria triple every 1/2 hour, how much bacteria in 3 hours
Set up the exponential function B(t) where t is the number of tripling times:
B(d) = 200 * (3^t)
3 hours = 6 (1/2 hour) periods, so we have 6 tripling times. We want to know B(6):
B(6) = 200 * (3^6)
B(6) = 200 * 729
B(6) = [B]145,800[/B]

If an employee starts saving with $750 and increases his savings by 8% each month, what will be his

If an employee starts saving with $750 and increases his savings by 8% each month, what will be his total savings after 10 months?
Set up the savings function S(m), where m is the number of months and I is the interest rate growth:
S(m) = Initial Amount * (1 + i)^m
Plugging in our number at m = 10 months we get:
S(10) = 750 * (1 + 0.08)^10
S(10) = 750 * 1.08^10
S(10) = [B]$1,619.19[/B]

If Bill's salary is $25 and he gets a 20¢ commission on every newspaper he sells, how many must he s

If Bill's salary is $25 and he gets a 20¢ commission on every newspaper he sells, how many must he sell to make $47
Set up bills Earnings function E(n) where n is the number of newspapers he sells:
E(n) =. Cost per newspaper * number of newspapers sold + base salary
E(n) = 0.2n + 25
We're asked to find n when E(n) = 47, so we set E(n) = 47 and solve for n:
0.2n + 25 = 47
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=0.2n%2B25%3D47&pl=Solve']equation solver[/URL], we get:
n = [B]110[/B]

If labor (x) costs $249 per unit, materials (y) cost $162 per unit, and capital (z) costs $ 7

If labor (x) costs $249 per unit, materials (y) cost $162 per unit, and capital (z) costs $ 77 per unit, write a function for total cost.
Total Cost = Labor Total Cost + Materials Total Cost + Capital Total Cost
Total Cost = [B]249x + 162y + 77z[/B]

If one calculator costs d dollars, what is the cost, in dollars, of 13 calculators?

If one calculator costs d dollars, what is the cost, in dollars, of 13 calculators?
Set up cost function C(n), where n is the number of calculators:
C(n) = dn
C(13) = [B]13d[/B]

If sin(26)=x what does cos(64) equal?

If sin(26)=x what does cos(64) equal?
Using our cofunction calculator, we see the cofunction of sin(26) = cos(64).
Therefore, sin(26) = cos(64), so cos(64) = [B]x[/B]

In 2010 a algebra book cost $125. In 2015 the book cost $205. Whats the linear function since 2010?

In 2010 a algebra book cost $125. In 2015 the book cost $205. Whats the linear function since 2010?
In 5 years, the book appreciated 205 - 125 = 80 in value.
80/5 = 16.
So each year, the book increases 16 in value. Set up the cost function:
[B]C(y) = 16y where y is the number of years since 2010[/B]

In 2016 the geese population was at 750. the geese population is expected to grow at a rate of 12% e

In 2016 the geese population was at 750. the geese population is expected to grow at a rate of 12% each year. What is the geese population in 2022?
12% is also 0.12. We have the population growth function:
P(t) = 750(1.12)^t
2022 - 2016 is 6 years of growth. We want P(6).
P(6) = 750(1.12)^6
P(6) = 750(1.9738)
[B]P(6) = 1,480.36 ~ 1,480[/B]

Isabel is making face mask. She spends $50 on supplies and plans on selling them for $4 per mask. Ho

Isabel is making face mask. She spends $50 on supplies and plans on selling them for $4 per mask. How many mask does have to make in order to make a profit equal to $90?
[U]Set up the cost function C(m) where m is the number of masks:[/U]
C(m) = supply cost
C(m) = 50
[U]Set up the cost function R(m) where m is the number of masks:[/U]
R(m) = Sale Price * m
R(m) = 4m
[U]Set up the profit function P(m) where m is the number of masks:[/U]
P(m) = R(m) - C(m)
P(m) = 4m - 50
The problems asks for profit of 90, so we set P(m) = 90:
4m - 50 = 90
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=4m-50%3D90&pl=Solve']type it in our search engine[/URL] and we get:
m = [B]35[/B]

It costs $2.50 to rent bowling shoes. Each game costs $2.25. You have $9.25. How many games can you

It costs $2.50 to rent bowling shoes. Each game costs $2.25. You have $9.25. How many games can you bowl. Writing an equation and give your answer.
Let the number of games be g. we have the function C(g):
C(g) = cost per game * g + bowling shoe rental
C(g) = 2.25g + 2.50
The problem asks for g when C(g) = 9.25
2.25g + 2.50 = 9.25
To solve this equation, we[URL='https://www.mathcelebrity.com/1unk.php?num=2.25g%2B2.50%3D9.25&pl=Solve'] type it in our search engine[/URL] and we get:
g = [B]3[/B]

it costs $75.00 for a service call from shearin heating and air conditioning company. the charge for

it costs $75.00 for a service call from shearin heating and air conditioning company. the charge for labor is $60.00 . how many full hours can they work on my air conditioning unit and still stay within my budget of $300.00 for repairs and service?
Our Cost Function is C(h), where h is the number of labor hours. We have:
C(h) = Variable Cost * Hours + Fixed Cost
C(h) = 60h + 75
Set C(h) = $300
60h + 75 = 300
[URL='https://www.mathcelebrity.com/1unk.php?num=60h%2B75%3D300&pl=Solve']Running this problem in the search engine[/URL], we get [B]h = 3.75[/B].

It costs a $20 flat fee to rent a lawn mower, plus $5 a day starting with the first day. Let x repre

It costs a $20 flat fee to rent a lawn mower, plus $5 a day starting with the first day. Let x represent the number of days rented, so y represents the charge to the user (in dollars)
Set up our function:
[B]y = 20 + 5x[/B]

James has a weekly allowance of 5 plus 1.50 for each chore c he does

James has a weekly allowance of 5 plus 1.50 for each chore c he does
We build the allowance function A(c) where c is each chore
A(c) = cost per chore * c + Weekly Allowance
Plugging in our numbers, we get:
[B]A(c) = 1.50c + 5[/B]

Jazmin is a hairdresser who rents a station in a salon for daily fee. The amount of money (m) Jazmin

Jazmin is a hairdresser who rents a station in a salon for daily fee. The amount of money (m) Jazmin makes from any number of haircuts (n) a day is described by the linear function m = 45n - 30
A) A haircut costs $30, and the station rent is $45
B) A haircut costs $45, and the station rent is $30.
C) Jazmin must do 30 haircuts to pay the $45 rental fee.
D) Jazmin deducts $30 from each $45 haircut for the station rent.
[B]Answer B, since rent is only due once. Profit is Revenue - Cost[/B]

Jessica tutors chemistry. For each hour that she tutors, she earns 30 dollars. Let E be her earnings

Jessica tutors chemistry. For each hour that she tutors, she earns 30 dollars. Let E be her earnings (in dollars) after tutoring for H hours. Write an equation relating E to H . Then use this equation to find Jessicas earnings after tutoring for 19 hours.
Set up a function of h hours for tutoring:
[B]E(h) = 30h[/B]
We need to find E(19)
E(19) = 30(19)
E(19) = [B]570[/B]

Jim has $440 in his savings account and adds $12 per week to the account. At the same time, Rhonda h

Jim has $440 in his savings account and adds $12 per week to the account. At the same time, Rhonda has $260 in her savings account and adds $18 per week to the account. How long will it take Rhonda to have the same amount in her account as Jim?
[U]Set up Jim's savings function S(w) where w is the number of weeks of savings:[/U]
S(w) = Savings per week * w + Initial Savings
S(w) = 12w + 440
[U]Set up Rhonda's savings function S(w) where w is the number of weeks of savings:[/U]
S(w) = Savings per week * w + Initial Savings
S(w) = 18w + 260
The problems asks for w where both savings functions equal each other:
12w + 440 = 18w + 260
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=12w%2B440%3D18w%2B260&pl=Solve']type this equation into our math engine[/URL] and we get:
w = [B]30[/B]

Joe opens a bank account that starts with $20 and deposits $10 each week. Bria has a different accou

Joe opens a bank account that starts with $20 and deposits $10 each week. Bria has a different account that starts with $1000 but withdraws $15 each week. When will Joe and Bria have the same amount of money?
Let w be the number of weeks. Deposits mean we add money and withdrawals mean we subtract money.
[U]Joe's Balance function B(w) where w is the number of weeks:[/U]
20 + 10w
[U]Bria's Balance function B(w) where w is the number of weeks:[/U]
1000 - 15w
[U]The problem asks for when both balances will be the same. So we set them equal to each other and solve for w:[/U]
20 + 10w = 1000 - 15w
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=20%2B10w%3D1000-15w&pl=Solve']type this equation into our search engine[/URL] and we get:
w = 39.2
We round up to full week and get:
w = [B]40[/B]

John is paid a retainer of $550 a week as well as a 2% commission on sales made. Find his income for

John is paid a retainer of $550 a week as well as a 2% commission on sales made. Find his income for the week if in one week he sells cars worth of $80000
Set up the income function C(s) where s is the number of sales for a week. Since 2% can be written as 0.02, we have:
I(s) = Retainer + 2% of sales
I(s) = 550 + 0.02s
The problem asks for a I(s) where s = 80,000:
I(s) = 550 + 0.02(80000)
I(s) = 550 + 1600
I(s) = [B]2150[/B]

Julia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages and

Julia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages and .10 cents for additional texts. How many texts did she send out?
Let m be the number of messages. We have a cost function of:
C(m) = 9 + 0.1(m - 600)
We are given C(m) = 18.20
18.20 = 9 + 0.1(m - 600)
18.20 = 9 + 0.1m - 60
Combine like terms:
18.20 = 0.1m - 51
Add 51 to each side
0.1m = 69.20
Divide each side by 0.1
[B]m = 692[/B]

Julio has $150. Each week, he saves an additional $10. Write a function f(x) that models the total a

Julio has $150. Each week, he saves an additional $10. Write a function f(x) that models the total amount of money Julio has after x weeks
f(x) = Savings per week * number of weeks + starting amount
f(x) = [B]10x + 150[/B]

Karen earns $20 per hour and already has $400 saved, and wants to save $1200. How many hours until b

Karen earns $20 per hour and already has $400 saved, and wants to save $1200. How many hours until bob gets his $1200 goal?
Set up he savings function S(h) where h is the number of hours needed:
S(h) = savings per hour * h + current savings amount
S(h) = 20h + 400
The question asks for h when S(h) = 1200:
20h + 400 = 1200
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=20h%2B400%3D1200&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]40[/B]

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car th

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car that keisha wants to buy costs at least $5440. How many hours does Keisha need to babysit to earn enough to buy the car
Set up the Earning function E(h) where h is the number of hours Keisha needs to babysit:
E(h) = 8h + 1300
The question asks for h when E(h) is at least 5440. The phrase [I]at least[/I] means an inequality, which is greater than or equal to. So we have:
8h + 1300 >= 5440
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=8h%2B1300%3E%3D5440&pl=Solve']type it in our search engine[/URL] and we get:
h >= [B]517.5[/B]

kim and jason just had business cards made. kim’s printing company charged a one time setup fee of $

kim and jason just had business cards made. kim’s printing company charged a one time setup fee of $8 and then $20 per box of cards. jason,meanwhile ordered his online. they cost $8 per box. there was no setup fee, but he had to pay $20 to have his order shipped to his house. by coincidence, kim and jason ended up spending the same amount on their business cards. how many boxes did each buy? how much did each spend?
Set up Kim's cost function C(b) where b is the number of boxes:
C(b) = Cost per box * number of cards + Setup Fee + Shipping Fee
C(b) = 20c + 8 + 0
Set up Jason's cost function C(b) where b is the number of boxes:
C(b) = Cost per box * number of cards + Setup Fee + Shipping Fee
C(b) = 8c + 0 + 20
Since Kim and Jason spent the same amount, set both cost equations equal to each other:
20c + 8 = 8c + 20
[URL='https://www.mathcelebrity.com/1unk.php?num=20c%2B8%3D8c%2B20&pl=Solve']Type this equation into our search engine[/URL] to solve for c, and we get:
c = 1
How much did they spend? We pick either Kim's or Jason's cost equation since they spent the same, and plug in c = 1:
Kim:
C(1) = 20(1) + 8
C(1) = 20 + 8
C(1) = [B]28
[/B]
Jason:
C(1) = 8(1) + 20
C(1) = 8 + 20
C(1) = [B]28[/B]

Kim earns $30 for babysitting on Friday nights. She makes an average of $1.25 in tips per hour. Writ

Kim earns $30 for babysitting on Friday nights. She makes an average of $1.25 in tips per hour. Write the function of Kim's earnings, and solve for how much she would make after 3 hours.
Set up the earnings equation E(h) where h is the number of hours. We have the function:
E(h) = 1.25h + 30
The problem asks for E(3):
E(3) = 1.25(3) + 30
E(3) = 4.75 + 30
E(3) = [B]$34.75[/B]

Last week at the business where you work, you sold 120 items. The business paid $1 per item and sol

Last week at the business where you work, you sold 120 items. The business paid $1 per item and sold them for $3 each. What profit did the business make from selling the 120 items?
Let n be the number of items. We have the following equations:
Cost Function C(n) = n
For n = 120, we have C(120) = 120
Revenue Function R(n) = 3n
For n = 120, we have R(120) = 3(120) = 360
Profit = Revenue - Cost
Profit = 360 - 120
Profit = [B]240[/B]

Leonard earned $100 from a bonus plus $15 per day (d) at his job this week. Which of the following e

Leonard earned $100 from a bonus plus $15 per day (d) at his job this week. Which of the following expressions best represents Leonards income for the week?
We set up an income function I(d), were d is the number of days Leonard works:
[B]I(d) = 15d + 100
[/B]
Each day, Leonard earns $15. Then we add on the $100 bonus

Lily needs an internet connectivity package for her firm. She has a choice between CIVISIN and GOMI

Lily needs an internet connectivity package for her firm. She has a choice between CIVISIN and GOMI with the following monthly billing policies. Each company's monthly billing policy has an initial operating fee and charge per megabyte.
Operating Fee charge per Mb
CIVSIN 29.95 0.14
GOMI 4.95 0.39
(i) Write down a system of equations to model the above situation
(ii) At how many Mb is the monthly cost the same? What is the equal monthly cost of the two plans?
(i) Set up a cost function C(m) for CIVSIN where m is the number of megabytes used:
C(m) = charge per Mb * m + Operating Fee
[B]C(m) = 0.14m + 29.95[/B]
Set up a cost function C(m) for GOMI where m is the number of megabytes used:
C(m) = charge per Mb * m + Operating Fee
[B]C(m) = 0.39m + 4.95
[/B]
(ii) At how many Mb is the monthly cost the same?
Set both cost functions equal to each other:
0.14m + 29.95 = 0.39m + 4.95
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.14m%2B29.95%3D0.39m%2B4.95&pl=Solve']type this equation into our search engine[/URL] and we get:
m = [B]100[/B]
(ii) What is the equal monthly cost of the two plans?
CIVSIN - We want C(100) from above where m = 100
C(100) = 0.14(100) + 29.95
C(100) = 14 + 29.95
C(100) = [B]43.95[/B]
GOMI - We want C(100) from above where m = 100
C(100) = 0.39(100) + 4.95
C(100) = 39 + 4.95
C(100) = [B]43.95[/B]

Marla wants to rent a bike Green Lake Park has an entrance fee of $8 and charges $2 per hour for bik

Marla wants to rent a bike Green Lake Park has an entrance fee of $8 and charges $2 per hour for bike Oak Park has an entrance fee of $2 and charges $5 per hour for bike rentals she wants to know how many hours are friend will make the costs equal
[U]Green Lake Park: Set up the cost function C(h) where h is the number of hours[/U]
C(h) = Hourly Rental Rate * h + Entrance Fee
C(h) = 2h + 8
[U]Oak Park: Set up the cost function C(h) where h is the number of hours[/U]
C(h) = Hourly Rental Rate * h + Entrance Fee
C(h) = 5h + 2
[U]Marla wants to know how many hours make the cost equal, so we set Green Lake Park's cost function equal to Oak Parks's cost function:[/U]
2h + 8 = 5h + 2
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=2h%2B8%3D5h%2B2&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]2[/B]

Mary owns a store that sells computers. Her profit in dollars is represented by the function P(x) =

Mary owns a store that sells computers. Her profit in dollars is represented by the function P(x) = x^3 - 22x^2 - 240x, where x is the number of computers sold. Mary hopes to make a profit of at least $10,000 by the time she sells 36 computers. Explain whether Mary will meet her goal. Justify your reasoning.
Calculate P(10):
P(10) = 10^3 - 22(10)^2 - 240(10)
P(10) = 1000 - 2200 - 2400
P(10) = -3600
Mary will [B]not[/B] meet her goal of making a profit of at least $10,000 when she sells 36 computers because her profit is in the negative.

Matthew's pay increases by 20% each month. If his first pay is $450, determine the amount of his pay

Matthew's pay increases by 20% each month. If his first pay is $450, determine the amount of his pay in month 5.
Let me be the number of months. We have a pay functionalists P(m) as:
P(m) = Initial Pay * (1 + Increase %/100)^m
With m = 5, initial pay = 450, and Increase % = 20, we have
P(5) = 450 * (1.2)^5
P(5) = 450 * 2.48832
P(5) = [B]1,119.74[/B]

Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many

Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many weeks will megan and connor have saved the same amount
[U]Set up the Balance function B(w) where w is the number of weeks for Megan:[/U]
B(w) = savings per week * w + Current Balance
B(w) = 5.50w + 50
[U]Set up the Balance function B(w) where w is the number of weeks for Connor:[/U]
B(w) = savings per week * w + Current Balance
B(w) = 7.75w + 18.50
The problem asks for w when both B(w) are equal. So we set both B(w) equations equal to each other:
5.50w + 50 = 7.75w + 18.50
To solve this equation for w, we[URL='https://www.mathcelebrity.com/1unk.php?num=5.50w%2B50%3D7.75w%2B18.50&pl=Solve'] type it in our search engine[/URL] and we get:
w = [B]14[/B]

Melissa runs a landscaping business. She has equipment and fuel expenses of $264 per month. If she c

Melissa runs a landscaping business. She has equipment and fuel expenses of $264 per month. If she charges $53 for each lawn, how many lawns must she service to make a profit of at $800 a month?
Melissa has a fixed cost of $264 per month in fuel. No variable cost is given. Our cost function is:
C(x) = Fixed Cost + Variable Cost. With variable cost of 0, we have:
C(x) = 264
The revenue per lawn is 53. So R(x) = 53x where x is the number of lawns.
Now, profit is Revenue - Cost. Our profit function is:
P(x) = 53x - 264
To make a profit of $800 per month, we set P(x) = 800.
53x - 264 = 800
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=53x-264%3D800&pl=Solve']equation solver[/URL], we get:
[B]x ~ 21 lawns[/B]

mike went to canalside with $40 to spend. he rented skates for $10 and paid $3 per hour to skate.wha

mike went to canalside with $40 to spend. he rented skates for $10 and paid $3 per hour to skate.what is the greatest number of hours Mike could have skated?
Let h be the number of hours of skating. We have the cost function C(h):
C(h) = Hourly skating rate * h + rental fee
C(h) = 3h + 10
The problem asks for h when C(h) = 40:
3h + 10 = 40
To solve this equation for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=3h%2B10%3D40&pl=Solve']type it in our search engine[/URL] and we get:
h = [B]10[/B]

Mr. Chris’s new app “Tick-Tock” is the hottest thing to hit the app store since...ever. It costs $5

Mr. Chris’s new app “Tick-Tock” is the hottest thing to hit the app store since...ever. It costs $5 to buy the app and then $2.99 for each month that you subscribe (a bargain!). How much would it cost to use the app for one year? Write an equation to model this using the variable “m” to represent the number of months that you use the app.
Set up the cost function C(m) where m is the number of months you subscribe:
C(m) = Monthly Subscription Fee * months + Purchase fee
[B]C(m) = 2.99m + 5[/B]

Mr. Demerath has a large collection of Hawaiian shirts. He currently has 42 Hawaiian shirts. He gets

Mr. Demerath has a large collection of Hawaiian shirts. He currently has 42 Hawaiian shirts. He gets 2 more every month. After how many months will Mr. Demerath have at least 65 Hawaiian shirts?
We set up the function H(m) where m is the number of months that goes by. Mr. Demerath's shirts are found by:
H(m) = 2m + 42
The problem asks for m when H(m) = 65. So we set H(m) = 65:
2m + 42 = 65
To solve this equation for m, we[URL='https://www.mathcelebrity.com/1unk.php?num=2m%2B42%3D65&pl=Solve'] type it in our search engine [/URL]and we get:
m = [B]11.5[/B]

Nia is trying to decide between two possible jobs. Job A pays $2000 a month with a 2% annual raise.

Nia is trying to decide between two possible jobs. Job A pays $2000 a month with a 2% annual raise. Job B pays 24,000 a year with a $500 annual raise. Write a function to represent the annual salary for Job A after x years. Write a function to represent the annual salary for Job B after x years. After how many years would Nia have a greater salary at Job A?
Nia Job A salary at time t: S(t)
$2,000 per month equals $24,000 per year.
So we have S(t) = 24,000(1.o2)^t
Nia Job B salary at time t: S(t)
$24,000 per year.
So we have S(t) = 24,000 + 500t
We want to know t when Job A salary is greater than Job B Salary:
24,000(1.o2)^t > 24,000 + 500t
Time | A | B
0 | 24000 | 24000
1 | 24480 | 24500
2 | 24969.6 | 25000
3 | 25468.99 | 25500
4 | 25978.37 | 26000
5 | 26497.94 | 26500
6 | 27027.9 | 27000
7 | 27568.46 | 27500
8 | 28119.83 | 28000
9 | 28682.22 | 28500
10 | 29255.87 | 29000
11 | 29840.98 | 29500
12 | 30437.8 | 30000
13 | 31046.56 | 30500

Nick is given $50 to spend on a vacation . He decides to spend $5 a day. Write an equation that show

Nick is given $50 to spend on a vacation . He decides to spend $5 a day. Write an equation that shows how much money Nick has after x amount of days.
Set up the function M(x) where M(x) is the amount of money after x days. Since spending means a decrease, we subtract to get:
[B]M(x) = 50 - 5x[/B]

Number Property

This calculator determines if an integer you entered has any of the following properties:

* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)

* Evil Numbers or Odious Numbers

* Perfect Numbers, Abundant Numbers, or Deficient Numbers

* Triangular Numbers

* Prime Numbers or Composite Numbers

* Automorphic (Curious)

* Undulating Numbers

* Square Numbers

* Cube Numbers

* Palindrome Numbers

* Repunit Numbers

* Apocalyptic Power

* Pentagonal

* Tetrahedral (Pyramidal)

* Narcissistic (Plus Perfect)

* Catalan

* Repunit

* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)

* Evil Numbers or Odious Numbers

* Perfect Numbers, Abundant Numbers, or Deficient Numbers

* Triangular Numbers

* Prime Numbers or Composite Numbers

* Automorphic (Curious)

* Undulating Numbers

* Square Numbers

* Cube Numbers

* Palindrome Numbers

* Repunit Numbers

* Apocalyptic Power

* Pentagonal

* Tetrahedral (Pyramidal)

* Narcissistic (Plus Perfect)

* Catalan

* Repunit

Oceanside Bike Rental Shop charges 16 dollars plus 6 dollars an hour for renting a bike. Mary paid 5

Oceanside Bike Rental Shop charges 16 dollars plus 6 dollars an hour for renting a bike. Mary paid 58 dollars to rent a bike. How many hours did she pay to have the bike checked out ?
Set up the cost function C(h) where h is the number of hours you rent the bike:
C(h) = Hourly rental cost * h + initial rental charge
C(h) = 6h + 16
Now the problem asks for h when C(h) = 58, so we set C(h) = 58:
6h + 16 = 58
To solve this equation for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=6h%2B16%3D58&pl=Solve']type it in our math engine[/URL] and we get:
h = [B]7 hours[/B]

p(x)=2x-5 find the domain

p(x)=2x-5 find the domain
Using our[URL='http://www.mathcelebrity.com/function-calculator.php?num=2x-5&pl=Calculate'] function calculator[/URL]:
[B]All real numbers[/B]

Penelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total s

Penelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total sales in dollars, xx, which can be represented by g(x)=215+0.035x. Owen is paid $242 per week plus 2.5% of his total sales in dollars, xx, which can be represented by f(x)=242+0.025x. Determine the value of xx, in dollars, that will make their weekly pay the same.
Set the pay functions of Owen and Penelope equal to each other:
215+0.035x = 242+0.025x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=215%2B0.035x%3D242%2B0.025x&pl=Solve']equation calculator[/URL], we get:
[B]x = 2700[/B]

please answer this word problem

Time 1, distance apart is 105 + 85 = 190
So every hour, the distance between them is 190 * t where t is the number of hours. Set up our distance function:
D(t) = 190t
We want D(t) = 494
190t = 494
Divide each side by 190
[B]t = 2.6 hours[/B]

Poisson Distribution

Calculates the probability of 3 separate events that follow a poisson distribution.

It calculates the probability of exactly k successes P(x = k)

No more than k successes P (x <= k)

Greater than k successes P(x >= k)

Each scenario also calculates the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

It calculates the probability of exactly k successes P(x = k)

No more than k successes P (x <= k)

Greater than k successes P(x >= k)

Each scenario also calculates the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

Rachel borrowed 8000 at a rate of 10.5%, compounded monthly. Assuming she makes no payments, how muc

Rachel borrowed 8000 at a rate of 10.5%, compounded monthly. Assuming she makes no payments, how much will she owe after 4 years?
[U]Convert annual amounts to monthly[/U]
4 years = 12 * 4 = 48 months
i = .105/12 = 0.00875 monthly
[U]Build our accumulation function A(t) where t is the time in months[/U]
A(48) = 8,000 * (1.00875)^48
A(48) = 8,000 * 1.5192
A(48) = [B]12,153.60
[/B]
[URL='http://www.mathcelebrity.com/compoundint.php?bal=8000&nval=48&int=10.5&pl=Monthly']You can also use the balance calculator[/URL]

Rafael is a software salesman. His base salary is $1900 , and he makes an additional $40 for every c

Rafael is a software salesman. His base salary is $1900 , and he makes an additional $40 for every copy of Math is Fun he sells. Let p represent his total pay (in dollars), and let c represent the number of copies of Math is Fun he sells. Write an equation relating to . Then use this equation to find his total pay if he sells 22 copies of Math is Fun.
We want a sales function p where c is the number of copies of Math is Fun
p = Price per sale * c + Base Salary
[B]p = 40c + 1900
[/B]
Now, we want to know Total pay if c = 22
p = 40(22) + 1900
p = 880 + 1900
p = [B]2780[/B]

Researchers in Antarctica discovered a warm sea current under the glacier that is causing the glacie

Researchers in Antarctica discovered a warm sea current under the glacier that is causing the glacier to melt. The ice shelf of the glacier had a thickness of approximately 450 m when it was first discovered. The thickness of the ice shelf is decreasing at an average rate if 0.06 m per day.
Which function can be used to find the thickness of the ice shelf in meters x days since the discovery?
We want to build an function I(x) where x is the number of days since the ice shelf discovery.
We start with 450 meters, and each day (x), the ice shelf loses 0.06m, which means we subtract this from 450.
[B]I(x) = 450 - 0.06x[/B]

Roberto owns a trucking company. He charges $50 hook up fee and $2 per mile. How much to tow your ca

Roberto owns a trucking company. He charges $50 hook up fee and $2 per mile. How much to tow your car: 1mile , 2miles , 10miles ?
The Cost Function C(m) where m is the number of miles is written as:
C(m) = 2m + 50
The problem asks for C(1), C(2), and C(10)
Calculate C(1)
C(1) = 2(1) + 50
C(1) = 2 + 50
C(1) = [B]52[/B]
Calculate C(2)
C(2) = 2(2) + 50
C(2) = 4 + 50
C(2) = [B]54[/B]
Calculate C(10)
C(10) = 2(10) + 50
C(10) = 20 + 50
C(10) = [B]70[/B]

sales 45,000 commission rate is 3.6% and salary is $275

sales 45,000 commission rate is 3.6% and salary is $275
Set up the commission function C(s) where s is the salary:
C(s) = Commission * s + salary
We're given: C(s) = 45,000, commission = 3.6%, which is 0.036 and salary = 275, so we have:
0.036s + 275 = 45000
To solve for s, we type this equation into our search engine and we get:
s = [B]1,242,361.11[/B]

Sam purchased n notebooks. They were 4 dollars each. Write an equation to represent the total cost c

Sam purchased n notebooks. They were 4 dollars each. Write an equation to represent the total cost c that Sam paid.
Cost Function is:
[B]c = 4n[/B]
Or, using n as a function variable, we write:
c(n) = 4n

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next week, Sarah is scheduled to work 8 hours at the daycare center. Which of the following inequalities represents the number of hours (h) that Sandra needs to work at the restaurant next week to earn at least $156 from these two jobs?
Set up Sarah's earnings function E(h) where h is the hours Sarah must work at the restaurant:
12h + 9(8) >= 156 <-- The phrase [I]at least[/I] means greater than or equal to, so we set this up as an inequality. Also, the daycare earnings are $9 per hour * 8 hours
Multiplying through and simplifying, we get:
12h + 72 >= 156
We [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B72%3E%3D156&pl=Solve']type this inequality into the search engine[/URL], and we get:
[B]h>=7[/B]

Savannah is a salesperson who sells computers at an electronics store. She makes a base pay of $90 e

Savannah is a salesperson who sells computers at an electronics store. She makes a base pay of $90 each day and is also paid a commission for each sale she makes. One day, Savannah sold 4 computers and was paid a total of $100. Write an equation for the function P(x), representing Savannah's total pay on a day on which she sells x computers.
If base pay is $90 per day, then the total commission Savannah made for selling 4 computers is:
Commission = Total Pay - Base Pay
Commission = 100 - 90
Commission = $10
Assuming the commission for each computer is equal, we need to find the commission per computer:
Commission per computer = Total Commission / Number of Computers Sold
Commission per computer = 10/4
Commission per computer = $2.50
Now, we build the Total pay function P(x):
Total Pay = Base Pay + Commission * Number of Computers sold
[B]P(x) = 90 + 2.5x[/B]

Scientists are studying a cell that divides in half every 15 minutes. How many cells will there by a

Scientists are studying a cell that divides in half every 15 minutes. How many cells will there by after 2.5 hours?
Divide 2.5 hours into 15 minute blocks.
2.5 hours = 2(60) + 0.5(60) minutes
2.5 hours = 120 + 30 minutes
2.5 hours = 150 minutes
Now determine the amount of 15 minute blocks
150 minutes/15 minutes = 10 blocks or divisions
[LIST]
[*]We start with 1 cell at time 0, and double it every 15 minutes
[*]We have A(0) = 1, we want A(10).
[*]Our accumulation function is A(t) = A(0) * 2^t
[/LIST]
A(10) = 1 * 2^10
A(10) = [B]1024[/B]

Sequences

Given a function a(n) and a count of sequential terms you want to expand (n), this calcuator will determine the first (n) terms of your sequence, {a_{1}, a_{2}, ..., a_{n}}

Sigmoid Function

Calculates the Sigmoid Function S(x) given an x value

Soda cans are sold in a local store for 50 cents each. The factory has $900 in fixed costs plus 25 c

Soda cans are sold in a local store for 50 cents each. The factory has $900 in fixed costs plus 25 cents of additional expense for each soda can made. Assuming all soda cans manufactured can be sold, find the break-even point.
Calculate the revenue function R(c) where s is the number of sodas sold:
R(s) = Sale Price * number of units sold
R(s) = 50s
Calculate the cost function C(s) where s is the number of sodas sold:
C(s) = Variable Cost * s + Fixed Cost
C(s) = 0.25s + 900
Our break-even point is found by setting R(s) = C(s):
0.25s + 900 = 50s
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.25s%2B900%3D50s&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]18.09[/B]

Sound travels about 340 m/s. The function d(t) = 340t give the distance d(t),in meters., that sound

Sound travels about 340 m/s. The function d(t) = 340t give the distance d(t),in meters., that sound travel in T seconds. How far goes sound traveling 59s?
What we want is d(59)
d(59) = 340m/s(59s) = [B]20,060m[/B]

Stacy sells art prints for $12 each. Her expenses are $2.50 per print, plus $38 for equipment. How m

Stacy sells art prints for $12 each. Her expenses are $2.50 per print, plus $38 for equipment. How many prints must she sell for her revenue to equal her expenses?
Let the art prints be p
Cost function is 38 + 2p
Revenue function is 12p
Set cost equal to revenue
12p = 38 + 2p
Subtract 2p from each side
10p = 38
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=10p%3D38&pl=Solve']equation calculator[/URL] gives us [B]p = 3.8[/B]

Standard Normal Distribution

Givena normal distribution z-score critical value, this will generate the probability. Uses the NORMSDIST Excel function.

Students stuff envelopes for extra money. Their initial cost to obtain the information for the job w

Students stuff envelopes for extra money. Their initial cost to obtain the information for the job was $140. Each envelope costs $0.02 and they get paid $0.03per envelope stuffed. Let x represent the number of envelopes stuffed. (a) Express the cost C as a function of x. (b) Express the revenue R as a function of x. (c) Determine analytically the value of x for which revenue equals cost.
a) Cost Function
[B]C(x) = 140 + 0.02x[/B]
b) Revenue Function
[B]R(x) = 0.03x[/B]
c) Set R(x) = C(x)
140 + 0.02x = 0.03x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=140%2B0.02x%3D0.03x&pl=Solve']equation solver[/URL], we get x = [B]14,000[/B]

Suppose a city's population is 740,000. If the population grows by 12,620 per year, find the populat

Suppose a city's population is 740,000. If the population grows by 12,620 per year, find the population of the city in 7 years
Set up the population function P(y) where y is the number of years since now:
P(y) = Current population + Growth per year * y
Plugging in our numbers at y = 7, we get:
P(7) = 740000 + 12620(7)
P(7) = 740000 + 88340
P(7) = [B]828,340[/B]

Suppose a city's population is 740,000. If the population grows by 12,620 per year, find the populat

Suppose a city's population is 740,000. If the population grows by 12,620 per year, find the population of the city in 7 years.
We setup the population function P(y) where y is the number of years of population growth, g is the growth per year, and P(0) is the original population.
P(y) = P(0) + gy
Plugging in our numbers of y = 7, g = 12,620, and P(0) = 740,000, we have:
P(7) = 740,000 + 12,620 * 7
P(7) = 740,000 + 88,340
P(7) = [B]828,340[/B]

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in ga

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 20 gallons of fuel, the airplane weighs 2012 pounds. When carrying 55 gallons of fuel, it weighs 2208 pounds. How much does the airplane weigh if it is carrying 65 gallons of fuel?
Linear functions are written in the form of one dependent variable and one independent variable. Using g as the number of gallons and W(g) as the weight, we have:
W(g) = gx + c where c is a constant
We are given:
[LIST]
[*]W(20) = 2012
[*]W(55) = 2208
[/LIST]
We want to know W(65)
Using our givens, we have:
W(20) = 20x + c = 2012
W(55) = 55x + c = 2208
Rearranging both equations, we have:
c = 2012 - 20x
c = 2208 - 55x
Set them both equal to each other:
2012 - 20x = 2208 - 55x
Add 55x to each side:
35x + 2012 = 2208
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=35x%2B2012%3D2208&pl=Solve']equation solver[/URL], we see that x is 5.6
Plugging x = 5.6 back into the first equation, we get:
c = 2012 - 20(5.6)
c = 2012 - 112
c = 2900
Now that we have all our pieces, find W(65)
W(65) = 65(5.6) + 2900
W(65) = 264 + 2900
W(65) = [B]3264[/B]

Susan makes and sells purses. The purses cost her $15 each to make, and she sells them for $30 each.

Susan makes and sells purses. The purses cost her $15 each to make, and she sells them for $30 each. This Saturday, she is renting a booth at a craft fair for $50. Write an equation that can be used to find the number of purses Susan must sell to make a profit of $295
Set up the cost function C(p) where p is the number of purses:
C(p) = Cost per purse * p + Booth Rental
C(p) = 15p + 50
Set up the revenue function R(p) where p is the number of purses:
R(p) = Sale price * p
R(p) = 30p
Set up the profit function which is R(p) - C(p) equal to 295
30p - (15p + 50) = 295
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=30p-%2815p%2B50%29%3D295&pl=Solve']we type it into our search engine[/URL] and we get:
p = [B]23[/B]

Terry recorded the temperature every hour from 8 AM to 1 PM. The temperature at 8 AM was 19?. The te

Terry recorded the temperature every hour from 8 AM to 1 PM. The temperature at 8 AM was 19?. The temperature dropped 4? every hour. What was the temperature at 1 PM? Group of answer choices 1 degree
Set up our temperature function T(h) where h is the number of hours since 8 AM:
T(h) = 19 - 4h <-- We subtract 4h since each hour, the temperature drops 4 degrees
The questions asks for the temperature at 1PM. We need to figure out how many hours pass since 8 AM:
8 AM to 12 PM is 4 hours
12 PM to 1 PM is 1 hour
Total time is 5 hours
So we want T(5):
T(5) = 19 - 4(5)
T(5) = 19 - 20
T(5) = [B]-1?[/B]

The blue star publishing company produces daily "Star news". It costs $1200 per day to operate regar

The blue star publishing company produces daily "Star news". It costs $1200 per day to operate regardless of whether any newspaper are published. It costs 0.20 to publish each newspaper. Each daily newspaper has $850 worth of advertising and each newspaper is sold for $.30. Find the number of newspaper required to be sold each day for the Blue Star company to 'break even'. I.e all costs are covered.
Build our cost function where n is the number of newspapers sold:
C(n) = 1200+ 0.2n
Now build the revenue function:
R(n) = 850 + 0.3n
Break even is where cost and revenue are equal, so set C(n) = R(n)
1200+ 0.2n = 850 + 0.3n
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=1200%2B0.2n%3D850%2B0.3n&pl=Solve']equation solver[/URL], we get:
[B]n = 3,500[/B]

The charge to rent a trailer is $30 for up to 2 hours plus $9 per additional hour or portion of an

The charge to rent a trailer is $30 for up to 2 hours plus $9 per additional hour or portion of an hour. Find the cost to rent a trailer for 2.4 hours, 3 hours, and 8.5 hours.
Set up the cost function C(h), where h is the number of hours to rent the trailer. We have, for any hours greater than 2:
C(h) = 30 + 9(h - 2)
Simplified, we have:
C(h) = 9h - 18 + 30
C(h) = 9h + 12
The question asks for C(2.4), C(3), and C(8.5)
[U]Find C(2.4)[/U]
C(2.4) = 9(2.4) + 12
C(2.4) = 21.6 + 12
C(2.4) = [B]33.6
[/B]
[U]Find C(3)[/U]
C(3) = 9(3) + 12
C(3) = 27 + 12
C(2.4) = [B][B]39[/B][/B]
[U]Find C(8.5)[/U]
C(8.5) = 9(8.5) + 12
C(8.5) = 76.5 + 12
C(8.5) = [B]88.5[/B]

The club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer ble

The club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer blender t years after its purchase. How long will it take the blender to depreciate completely?
Complete depreciation means the salvage value is 0.
So S(t) = 0. We need to find t to make S(t) = 0
-4,500t + 54,000 = 0
Subtract 54,000 from each side
-4,500t = -54,000
Divide each side by -4,500
[B]t = 12[/B]

The cost of a taxi ride is $1.2 for the first mile and $0.85 for each additional mile or part thereo

The cost of a taxi ride is $1.2 for the first mile and $0.85 for each additional mile or part thereof. Find the maximum distance we can ride if we have $20.75.
We set up the cost function C(m) where m is the number of miles:
C(m) = Cost per mile after first mile * m + Cost of first mile
C(m) = 0.8(m - 1) + 1.2
C(m) = 0.8m - 0.8 + 1.2
C(m) = 0.8m - 0.4
We want to know m when C(m) = 20.75
0.8m - 0.4 = 20.75
[URL='https://www.mathcelebrity.com/1unk.php?num=0.8m-0.4%3D20.75&pl=Solve']Typing this equation into our math engine[/URL], we get:
m = 26.4375
The maximum distance we can ride in full miles is [B]26 miles[/B]

The cost of hiring a car for a day is $60 plus 0.25 cents per kilometer. Michelle travels 750 kilome

The cost of hiring a car for a day is $60 plus 0.25 cents per kilometer. Michelle travels 750 kilometers. What is her total cost
Set up the cost function C(k) where k is the number of kilometers traveled:
C(k) = 60 + 0.25k
The problem asks for C(750)
C(750) = 60 + 0.25(750)
C(750) = 60 + 187.5
C(750) = [B]247.5[/B]

the cost of x concert tickets if one concert ticket costs $97

the cost of x concert tickets if one concert ticket costs $97
The cost function C(x), where x is the number of concert tickets is:
[B]C(x) = 97x[/B]

The cost of x ice cream if one ice cream cost $9 and the fixed cost is $8142

The cost of x ice cream if one ice cream cost $9 and the fixed cost is $8142
Cost function is C(x) is:
C(x) = Cost per ice cream * number of ice creams + Fixed Cost
C(x) = [B]9x + 8142[/B]

the cost of x pounds of pork at $4.10 a pound

the cost of x pounds of pork at $4.10 a pound
Set up the cost function for pounds of pork:
[B]C(x) = 4.10x[/B]

The cost of x textbooks if one textbook costs $140

The cost of x textbooks if one textbook costs $140.
Set up a cost function where x is the number of textbooks:
[B]C(x) = 140x[/B]

The cost to rent a boat is $10. There is also charge of $2 for each person. Which expresion represen

The cost to rent a boat is $10. There is also charge of $2 for each person. Which expresion represents the total cost to rent a boat for p persons?
The cost function includes a fixed cost of $10 plus a variable cost of 2 persons for p persons:
[B]C(p) = 2p + 10[/B]

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number o

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed 1650 per day?
Set up the cost function where h is the number of hours:
C(h) = 150h + 450
We want C(h) <= 1650:
150h + 450 <= 1650
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=150h%2B450%3C%3D1650&pl=Solve']equation/inequality solver[/URL], we get:
[B]h <= 8[/B]

The dance committee of pine bluff middle school earns $72 from a bake sale and will earn $4 for each

The dance committee of pine bluff middle school earns $72 from a bake sale and will earn $4 for each ticket sold they sell to the Spring Fling dance. The dance will cost $400
Let t be the number of tickets sold. We have a Revenue function R(t):
R(t) = 4t + 72
We want to know t such that R(t) = 400. So we set R(t) = 400:
4t + 72 = 400
[URL='https://www.mathcelebrity.com/1unk.php?num=4t%2B72%3D400&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]t = 82[/B]

The enrollment at High School R has been increasing by 20 students per year. High School R currently

The enrollment at High School R has been increasing by 20 students per year. High School R currently has 200 students. High School T has 400 students and is decreasing 30 students per year. When will the two school have the same enrollment of students?
Set up the Enrollment function E(y) where y is the number of years.
[U]High School R:[/U]
[I]Increasing[/I] means we add
E(y) = 200 + 20y
[U]High School T:[/U]
[I]Decreasing[/I] means we subtract
E(y) = 400 - 30y
When the two schools have the same enrollment, we set the E(y) functions equal to each other
200 + 20y = 400 - 30y
To solve this equation for y, we [URL='https://www.mathcelebrity.com/1unk.php?num=200%2B20y%3D400-30y&pl=Solve']type it in our search engine[/URL] and we get:
y = [B]4[/B]

the fuel tank of a jet used gas at a constant rate of 300 gallons for each hour of flight. the tank

the fuel tank of a jet used gas at a constant rate of 300 gallons for each hour of flight. the tank can hold a maximum of 2400 gallons of gas. write an equation representing the amount of fuel left in the tank as a function of the number of hours spent flying.
We have an equation F(h) where h is the number of hours since the flight took off:
[B]F(h) = 2400 - 300h[/B]

The function f(x) = e^x(x - 3) has a critical point at x =

The function f(x) = e^x(x - 3) has a critical point at x =
The critical point is where the derivative equals 0.
We multiply through for f(x) to get:
f(x) = xe^x - 3e^x
Using the product rule on the first term f'g + fg', we get:
f'(x) = xe^x + e^x - 3e^x
f'(x) = xe^x -2e^x
f'(x) = e^x(x - 2)
We want f'(x) = 0
e^x(x - 2) = 0
When [B]x = 2[/B], then f'(x) = 0

The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?

The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?
f'(x) = 3x^2 - 48
Set this equal to 0:
3x^2 - 48 = 0
Add 48 to each side:
3x^2 = 48
Divide each side by 3:
x^2 = 16
Therefore, x = -4, 4
Test f(4)
f(4) = 4^3 - 48(4)
f(4) = 64 - 192
f(4) = [B]-128 <-- Local minimum[/B]
Test f(-4)
f(-4) = -4^3 - 48(-4)
f(-4) = -64 + 192
f(-4) = [B]128 <-- Local maximum[/B]

The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the ba

The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?
Take the [URL='http://www.mathcelebrity.com/dfii.php?term1=-30x%5E2+%2B+360x+%2B+785&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']derivative of the profit function[/URL]:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
[B]x = 6 <-- This is the ticket price to maximize profit[/B]
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
[B]P(6) = 1865[/B]

the output is double the input

the output is double the input
Double means multiply by 2. So this means a function with input of x and output of y such that:
[B]y = 2x[/B]

The polynomial function P(x) = 75x - 87,000 models the relationship between the number of computer

The polynomial function P(x) = 75x - 87,000 models the relationship between the number of computer briefcases x that a company sells and the profit the company makes, P(x). Find P (4000), the profit from selling 4000 computer briefcases.
Plug in 4,000 for x:
P(4000) = 75(4000) - 87,000
P(4000) = 300,000- 87,000
P(4000) = [B]213,000[/B]

The price p of a gym’s membership is $30 for an enrollment fee and $12 per week w to be a member. W

The price p of a gym’s membership is $30 for an enrollment fee and $12 per week w to be a member. What is the cost to be a member for 5 weeks?
Set up the cost function C(w)
C(w) = 12w + 30
The problem asks for C(5)
C(5) = 12(5) + 30
C(5) = 60 + 30
C(5) = [B]90[/B]

The revenue for selling x candles is given by f(x)=12x. The teams profit is $40 less than 80% of the

The revenue for selling x candles is given by f(x)=12x. The teams profit is $40 less than 80% of the revenue of selling x candles. write a function g to model the profit.
Profit = Revenue - Cost
We are given the revenue function f(x) = 12x. We are told the profit is 0.8(Revenue) - 40. Our profit function P(x) is:
P(x) = 0.8(12x) - 40
Simplifying, we have:
[B]P(x) = 9.6x - 40[/B]

The school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells for

The school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells for $40. Write a cost and revenue function and determine the break-even point.
[U]Calculate cost function C(b) with b as the number of books:[/U]
C(b) = Cost per book * b + Overhead
[B]C(b) = 15b + 5500[/B]
[U]Calculate Revenue Function R(b) with b as the number of books:[/U]
R(b) = Sales Price per book * b
[B]R(b) = 40b[/B]
[U]Calculate break even function E(b):[/U]
Break-even Point = Revenue - Cost
Break-even Point = R(b) - C(b)
Break-even Point = 40b - 15b - 5500
Break-even Point = 25b - 5500
[U]Calculate break even point:[/U]
Break-even point is where E(b) = 0. So we set 25b - 5500 equal to 0
25b - 5500 = 0
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=25b-5500%3D0&pl=Solve']type this equation into our search engine[/URL] and we get:
[B]b = 220[/B]

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer
Let the integer be x.
[LIST]
[*]The sum of the integer and 12 is written as x + 12.
[*]The square of a positive integer is written as x^2.
[/LIST]
We set these equal to each other:
x^2 = x + 12
Subtract x + 12 from each side:
x^2 - x - 12 = 0
We have a quadratic function. [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-x-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Run it through our search engine[/URL] and we get x = 3 and x = -4.
The problem asks for a positive integer, so we have [B]x = 3[/B]

The store is selling apples for $0.49 per pound. Write a function to model the cost of "p" pounds of

The store is selling apples for $0.49 per pound. Write a function to model the cost of "p" pounds of apples.
Let p be the pounds of apples. Our cost function is:
[B]C(p) = 0.49p[/B]

The total cost for 9 bracelets, including shipping was $72. The shipping charge was $9. Define your

The total cost for 9 bracelets, including shipping was $72. The shipping charge was $9. Define your variable and write an equation that models the cost of each bracelet.
We set up a cost function as fixed cost plus total cost. Fixed cost is the shipping charge of $9. So we have the following cost function where n is the cost of the bracelets:
C(b) = nb + 9
We are given C(9) = 72 and b = 9
9n + 9 = 72
[URL='https://www.mathcelebrity.com/1unk.php?num=9n%2B9%3D72&pl=Solve']Run this through our equation calculator[/URL], and we get [B]n = 7[/B].

The total cost of producing x units for which the fixed costs are $2900 and the cost per unit is $25

The total cost of producing x units for which the fixed costs are $2900 and the cost per unit is $25
Set up the cost function per x units:
[B]C(x) = 2900 + 25x[/B]

The total cost to fix your bike is $45 the parts cost $10 and the labor cost seven dollars per hour

The total cost to fix your bike is $45 the parts cost $10 and the labor cost seven dollars per hour how many hours were there:
Set up a cost function where h is the number of hours:
7h + 10 = 45
To solve for h, we t[URL='https://www.mathcelebrity.com/1unk.php?num=7h%2B10%3D45&pl=Solve']ype this equation into our search engine[/URL] and we get:
h = [B]5[/B]

There were 286,200 graphic designer jobs in a country in 2010. It has been projected that there will

There were 286,200 graphic designer jobs in a country in 2010. It has been projected that there will be 312,500 graphic designer jobs in 2020. (a) Using the data, find the number of graphic designer jobs as a linear function of the year.
[B][U]Figure out the linear change from 2010 to 2020[/U][/B]
Number of years = 2020 - 2010
Number of years = 10
[B][U]Figure out the number of graphic designer job increases:[/U][/B]
Number of graphic designer job increases = 312,500 - 286,200
Number of graphic designer job increases = 26,300
[B][U]Figure out the number of graphic designer jobs added per year[/U][/B]
Graphic designer jobs added per year = Total Number of Graphic Designer jobs added / Number of Years
Graphic designer jobs added per year = 26,300 / 10
Graphic designer jobs added per year = 2,630
[U][B]Build the linear function for graphic designer jobs G(y) where y is the year:[/B][/U]
G(y) = 286,200 + 2,630(y - 2010)
[B][U]Multiply through and simplify:[/U][/B]
G(y) = 286,200 + 2,630(y - 2010)
G(y) = 286,200 + 2,630y - 5,286,300
[B]G(y) = 2,630y - 5,000,100[/B]

To be a member of world fitness gym, it costs $60 flat fee and $30 per month. Maria has paid a total

To be a member of world fitness gym, it costs $60 flat fee and $30 per month. Maria has paid a total of $210 for her gym membership so far. How long has Maria been a member to the gym?
The cost function C(m) where m is the number of months for the gym membership is:
C(m) = 30m + 60
We're given that C(m) = 210 for Maria. We want to know the number of months (m) that Maria has been a member.
With C(m) = 210, we have:
30m + 60 =210
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=30m%2B60%3D210&pl=Solve']we type it in our search engine[/URL] and we get:
m = [B]5[/B]

To ship a package with UPS, the cost will be $7 for the first pound and $0.20 for each additional po

To ship a package with UPS, the cost will be $7 for the first pound and $0.20 for each additional pound. To ship a package with FedEx, the cost will be $5 for the first pound and $0.30 for each additional pound. How many pounds will it take for UPS and FedEx to cost the same? If you needed to ship a package that weighs 8 lbs, which shipping company would you choose and how much would you pay?
[U]UPS: Set up the cost function C(p) where p is the number of pounds:[/U]
C(p) = Number of pounds over 1 * cost per pounds + first pound
C(p) = 0.2(p - 1) + 7
[U]FedEx: Set up the cost function C(p) where p is the number of pounds:[/U]
C(p) = Number of pounds over 1 * cost per pounds + first pound
C(p) = 0.3(p - 1) + 5
[U]When will the costs equal each other? Set the cost functions equal to each other:[/U]
0.2(p - 1) + 7 = 0.3(p - 1) + 5
0.2p - 0.2 + 7 = 0.3p - 0.3 + 5
0.2p + 6.8 = 0.3p + 4.7
To solve this equation for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.2p%2B6.8%3D0.3p%2B4.7&pl=Solve']type it in our search engine[/URL] and we get:
p = [B]21
So at 21 pounds, both UPS and FedEx costs are equal
[/B]
Now, find out which shipping company has a better rate at 8 pounds:
[U]UPS:[/U]
C(8) = 0.2(8 - 1) + 7
C(8) = 0.2(7) + 7
C(8) = 1.4 + 7
C(8) = 8.4
[U]FedEx:[/U]
C(8) = 0.3(8 - 1) + 5
C(8) = 0.3(7) + 5
C(8) = 2.1 + 5
C(8) = [B]7.1[/B]
[B]Therefore, FedEx is the better cost at 8 pounds since the cost is lower[/B]
[B][/B]

Today a car is valued at $42000. the value is expected to decrease at a rate of 8% each year. what i

Today a car is valued at $42000. the value is expected to decrease at a rate of 8% each year. what is the value of the car expected to be 6 years from now.
Depreciation at 8% per year means it retains (100% - 8%) = 92% of it's value. We set up our depreciation function D(t), where t is the number of years from right now.
D(t) = $42,000(0.92)^t
The problem asks for D(6):
D(6) = $42,000(0.92)^6
D(6) = $42,000(0.606355)
D(6) = [B]$25,466.91[/B]

Trig Measurement

Given an angle θ, this calculates the following measurements:

Sin(θ) = Sine

Cos(θ) = Cosine

Tan(θ) = Tangent

Csc(θ) = Cosecant

Sec(θ) = Secant

Cot(θ) = Cotangent

Arcsin(x) = θ = Arcsine

Arccos(x) = θ = Arccosine

Arctan(x) =θ = Arctangent

Also converts between Degrees and Radians and Gradians

Coterminal Angles as well as determine if it is acute, obtuse, or right angle. For acute angles, a cofunction will be determined. Also shows the trigonometry function unit circle

Sin(θ) = Sine

Cos(θ) = Cosine

Tan(θ) = Tangent

Csc(θ) = Cosecant

Sec(θ) = Secant

Cot(θ) = Cotangent

Arcsin(x) = θ = Arcsine

Arccos(x) = θ = Arccosine

Arctan(x) =θ = Arctangent

Also converts between Degrees and Radians and Gradians

Coterminal Angles as well as determine if it is acute, obtuse, or right angle. For acute angles, a cofunction will be determined. Also shows the trigonometry function unit circle

Uniform Distribution

This calculates the following items for a uniform distribution

* Probability Density Function (PDF) ƒ(x)

* Cumulative Distribution Function (CDF) F(x)

* Mean, Variance, and Standard Deviation

Calculates moment number t using the moment generating function

* Probability Density Function (PDF) ƒ(x)

* Cumulative Distribution Function (CDF) F(x)

* Mean, Variance, and Standard Deviation

Calculates moment number t using the moment generating function

Use the definite integral to find the area between the x-axis and the function f(x)= x^2-x-12 over t

Use the definite integral to find the area between the x-axis and the function f(x)= x^2-x-12 over the interval [ -5, 10].
Using our [URL='http://www.mathcelebrity.com/dfii.php?term1=x%5E2-x-12&fpt=0&ptarget1=0&ptarget2=0&itarget=-5%2C10&starget=0%2C1&nsimp=8&pl=Integral']integral calculator[/URL], we get:
[B]157.5[/B]

What does y=f(x) mean

What does y=f(x) mean
It means y = a function of the variable x.
x is the independent variable and y is the dependent variable.
f(x) means a function in terms of x

What is a Function

This lesson walks you through what a function is, how to write a function, the part of a function, and how to evaluate the outputs of a function.

This lesson also shows you the domain and range of a function. This lesson shows you the y-intercept of a function and the x-intercept of a function. Also shows Relation and function

This lesson also shows you the domain and range of a function. This lesson shows you the y-intercept of a function and the x-intercept of a function. Also shows Relation and function

Write a system of equations to describe the situation below, solve using any method, and fill in the

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Hugo is going to send some flowers to his wife. Somerville Florist charges $2 per rose, plus $39 for the vase. Dwaynes Flowers, in contrast, charges $3 per rose and $10 for the vase. If Hugo orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. What would the total cost be? How many roses would there be?
Let r be the number of roses and C(r) be the cost function. The vase is a one-time cost.
Somerville Florist:
C(r) = 2r + 39
Dwaynes Flowers
C(r) = 3r + 10
Set them equal to each other:
2r + 39 = 3r + 10
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2r%2B39%3D3r%2B10&pl=Solve']equation calculator[/URL], we get:
[B]r = 29[/B]

Xavier has $132 to buy a video game. Each game costs $12. Write an equation to find the number of ga

Xavier has $132 to buy a video game. Each game costs $12. Write an equation to find the number of games Xavier can purchase.
Let g be the number of games, we have a cost function C(g)
C(g) = 12g
We want to find g such that C(g) = 132
12g = 132
Divide each side by 12
[B]g = 11[/B]

You and a friend want to start a business and design t-shirts. You decide to sell your shirts for $1

You and a friend want to start a business and design t-shirts. You decide to sell your shirts for $15 each and you paid $6.50 a piece plus a $50 set-up fee and $25 for shipping. How many shirts do you have to sell to break even? Round to the nearest whole number.
[U]Step 1: Calculate Your Cost Function C(s) where s is the number of t-shirts[/U]
C(s) = Cost per Shirt * (s) Shirts + Set-up Fee + Shipping
C(s) = $6.50s + $50 + $25
C(s) = $6.50s + 75
[U]Step 2: Calculate Your Revenue Function R(s) where s is the number of t-shirts[/U]
R(s) = Price Per Shirt * (s) Shirts
R(s) = $15s
[U]Step 3: Calculate Break-Even Point[/U]
Break Even is where Cost = Revenue. Set C(s) = R(s)
$6.50s + 75 = $15s
[U]Step 4: Subtract 6.5s from each side[/U]
8.50s = 75
[U]Step 5: Solve for s[/U]
[URL='https://www.mathcelebrity.com/1unk.php?num=8.50s%3D75&pl=Solve']Run this through our equation calculator[/URL] to get s = 8.824. We round up to the next integer to get [B]s = 9[/B].
[B][URL='https://www.facebook.com/MathCelebrity/videos/10156751976078291/']FB Live Session[/URL][/B]

You deposit $750 in an account that earns 5% interest compounded quarterly. Show and solve a functio

You deposit $750 in an account that earns 5% interest compounded quarterly. Show and solve a function that represents the balance after 4 years.
The Accumulated Value (A) of a Balance B, with an interest rate per compounding period (i) for n periods is:
A = B(1 + i)^n
[U]Givens[/U]
[LIST]
[*]4 years of quarters = 4 * 4 = 16 quarters. So this is t.
[*]Interest per quarter = 5/4 = 1.25%
[*]Initial Balance (B) = 750.
[/LIST]
Using our [URL='https://www.mathcelebrity.com/compoundint.php?bal=750&nval=16&int=5&pl=Quarterly']compound balance interest calculator[/URL], we get the accumulated value A:
[B]$914.92[/B]

You have read a 247 page book for a class and decide to read 18 pages a night. How many pages are le

You have read a 247 page book for a class and decide to read 18 pages a night. How many pages are left in the book if you have been reading for n nights?
Set up the remaining pages read function R(n). We have:
[B]R(n) = 247 - 18n[/B]

You open a hat stand in the mall with an initial start-up cost of $1500 plus 50 cents for every hat

You open a hat stand in the mall with an initial start-up cost of $1500 plus 50 cents for every hat you stock your booth with. a) What is your cost function?
Set up the cost function C(h) where h is the number of hats you stock:
C(h) = Cost per hat * h hats + Start Up Cost
[B]C(h) = 0.5h + 1500[/B]

You open up a savings account. Your initial deposit is $300. You plan to add in $50 per month to sav

You open up a savings account. Your initial deposit is $300. You plan to add in $50 per month to save up for college. Write an equation to represent the situation.
Let m be the number of months. We have a Savings account function S(m):
S(m) = Monthly deposit * number of months + Initial Deposit
[B]S(m) = 50m + 300[/B]

You pay 510.00 to rent a storage unit for 3 months the total cost includes an initial deposit plus a

You pay 510.00 to rent a storage unit for 3 months the total cost includes an initial deposit plus a monthly fee of 160.00. Write and equation that represents your total cost Y in dollars after X months.
Set up the cost function Y where x is the number of months you rent
[B]Y = 160x + 510[/B]

You purchase a new car for $35,000. The value of the car depreciates at a rate of 8.5% per year. If

You purchase a new car for $35,000. The value of the car depreciates at a rate of 8.5% per year. If the rate of decrease continues, what is the value of your car in 5 years?
Set up the depreciation function D(t), where t is the time in years from purchase. We have:
D(t) = 35,000(1 - 0.085)^t
Simplified, a decrease of 8.5% means it retains 91.5% of it's value each year, so we have:
D(t) = 35,000(0.915)^t
The problem asks for D(5)
D(5) = 35,000(0.915)^5
D(5) = 35,000(0.64136531607)
D(5) = $[B]22,447.79[/B]

You rent skates for $5 and pay $1 an hour for skating per person. Write an equation.

You rent skates for $5 and pay $1 an hour for skating per person. Write an equation.
Let the number of hours be h. Our cost function C(h) is:
C(h) = Cost per hour * hourly rate + rental fee
Plugging in our numbers, we get:
[B]C(h) = h + 5[/B]

you start with 150$ in year bank account if you save $28 a year with equation would model your savin

you start with 150$ in year bank account if you save $28 a year with equation would model your savings find equation.
We create a savings function S(y) where y is the number of years since the start.
S(y) = Savings per year * y + initial savings
[B]S(y) = 28y + 150[/B]

You started this year with $491 saved and you continue to save an additional $11 per month. Write an

You started this year with $491 saved and you continue to save an additional $11 per month. Write an algebraic expression to represent the total amount of money saved after m months.
Set up a savings function for m months
[B]S(m) = 491 + 11m[/B]

You work for a remote manufacturing plant and have been asked to provide some data about the cost of

You work for a remote manufacturing plant and have been asked to provide some data about the cost of specific amounts of remote each remote, r, costs $3 to make, in addition to $2000 for labor. Write an expression to represent the total cost of manufacturing a remote. Then, use the expression to answer the following question. What is the cost of producing 2000 remote controls?
We've got 2 questions here.
Question 1: We want the cost function C(r) where r is the number of remotes:
C(r) = Variable Cost per unit * r units + Fixed Cost (labor)
[B]C(r) = 3r + 2000
[/B]
Question 2: What is the cost of producing 2000 remote controls.
In this case, r = 2000, so we want C(2000)
C(2000) = 3(2000) + 2000
C(2000) = 6000 + 2000
C(2000) = [B]$8000[/B]

Your clothes washer stopped working during the spin cycle and you need to get a person in to fix the

Your clothes washer stopped working during the spin cycle and you need to get a person in to fix the washer. Company A costs $20 for the visit and $15 for every hour the person is there to fix the problem. Company B costs $40 for the visit and $5 for every hour the person is there to fix the problem. When would Company B be cheaper than Company A?
Set up the cost functions:
[LIST]
[*]Company A: C(h) = 15h + 20
[*]Company B: C(h) = 5h + 40
[/LIST]
Set them equal to each other:
15h + 20 = 5h + 40
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=15h%2B20%3D5h%2B40&pl=Solve']equation solver[/URL], we get h = 2.
With [B]h = 3[/B] and beyond, Company B becomes cheaper than Company A.