223 results

square - a plane figure (regular quadrilateral) with four equal straight sides and four right angles

(Sqrt(24) + sqrt(96))/2

[SIZE=6][B](Sqrt(24) + sqrt(96))/2[/B]
[B][/B]
[B]Simplify sqrt(24)[/B]
24 = 6 * 4 where 4 is the perfect square
sqrt(24) = 2 * sqrt(6)
[B]Simplify sqrt(96)[/B]
96 = 16 * 6 where 16 is the perfect square
sqrt(96) = 4 * sqrt(6)
Simplified, we have:
(2 * sqrt(6) + 4 * sqrt(6))/2
6 * sqrt(6)/2
[B]3 * sqrt(6)[/B][/SIZE]
[SIZE=6]
[/SIZE]
[MEDIA=youtube]h-4eZOFUR4I[/MEDIA]

-x squared

-x squared
We take -x and raise it to the 2nd power:
(-x)^2 = -x * -x = [B]x^2[/B]

1 multiplied by b squared multiplied by c squared

1 multiplied by b squared multiplied by c squared
b squared means we raise b to the power of 2:
b^2
c squared means we raise c to the power of 2:
c^2
b squared multiplied by c squared
b^2c^2
1 multiplied by b squared multiplied by c squared means we multiply 1 by b^2c^2
1b^2c^2
Multiplying by 1 can be written by [U][I]removing[/I][/U] the 1 since it's an identity multiplication:
[B]b^2c^2[/B]

1, 9, 25, 49, .......... What is next

1, 9, 25, 49, .......... What is next
1^2 = 1
3^2 = 9
5^2 = 25
7^2 = 49
So this pattern takes odd numbers and squares them. Our next odd number is 9:
9^2 = [B]81[/B]

1/3c increased by the square root of d

1/3c increased by the square root of d
square root of d:
sqrt(d)
1/3c increased by the square root of d
[B]1/3c + sqrt(d)[/B]

1/n^2 = 3/192

1/n^2 = 3/192
Cross multiply:
192 * 1 = 3 * n^2
3n^2 = 192
Divide each side by 3:
3n^2/3 = 192/3
Cancel the 3's on the left side:
n^2 = 64
Take the square root of both sides:
n = [B]8 or -8[/B]

10 times the square of a number w divided by 12

10 times the square of a number w divided by 12
The square of a number w
w^2
10 times this
10w^2
Divided by 12
[B]10w^2/12[/B]

15 less than a number squared

15 less than a number squared
A number is denoted by an arbitrary variable, let's call it x.
x
Squared means we raise that number to a power of 2
x^2
15 less means we subtract
[B]x^2 -15[/B]

2 times b squared minus 6

2 times b squared minus 6
b squared means we raise b to the 2nd power:
b^2
2 times b squared
2b^2
Minus 6:
[B]2b^2 - 6[/B]

2 times x squared minus 4 times x

2 times x squared minus 4 times x
2x^2 - 4x

2x plus 8, quantity squared

2x plus 8, quantity squared
2x plus 8 means we add 8 to 2x:
2x + 8
Squaring the quantity means we raise it to the power of 2:
[B](2x + 8)^2[/B]

3 is subtracted from square of a number

3 is subtracted from square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Square of a number means we raise x to the 2nd power:
x^2
3 is subtracted from square of a number
[B]x^2 - 3[/B]

3 is subtracted from the square of x

3 is subtracted from the square of x
Let's take this algebraic expression in two parts:
Part 1: The square of x means we raise x to the power of 2:
x^2
Part 2: 3 is subtracted means we subtract 3 from x^2
[B]x^2 - 3[/B]

3 times the square of a number x minus 12

3 times the square of a number x minus 12.
Build the algebraic expression piece by piece:
[LIST]
[*]Square of a number x: x^2
[*]3 times this: 3x^2
[*]Minus 12: [B]3x^2 - 12[/B]
[/LIST]

3 times x squared minus 4 times x

3 times x squared minus 4 times x
[U]x squared[/U]
x^2
[U]3 times x squared:[/U]
3x^2
[U]4 times x:[/U]
4x
[U]3 times x squared minus 4 times x[/U]
[B]3x^2 - 4x[/B]

30 increased by 3 times the square of a number

Let "a number" equal the arbitrary variable x.
The square of that is x^2.
3 times the square of that is 3x^2.
Now, 30 increased by means we add 3x^2 to 30
30 + 3x^2

30 increased by 3 times the square of a number

30 increased by 3 times the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
The square of a number means we raise x to the power of 2:
x^2
3 times the square:
3x^2
The phrase [I]increased by[/I] means we add 3x^2 to 30:
[B]30 + 3x^2[/B]

4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section

4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section of a picture frame. Determine the area inside the wooden frame.
Area inside forms a square, with a length of 30 - 3 - 3 = 24. We subtract 3 twice, because we account for 2 rectangular strips with a width of 3.
Area of a square is side * side. So we have 24 * 24 = [B]576cm^2[/B]

4 times the sum of 3 plus x squared

4 times the sum of 3 plus x squared
x squared means we raise x to the power of 2:
x^2
3 plus x squared:
3 + x^2
4 times the sum of 3 plus x squared
3(3 + x^2)

5 squared minus a number x

5 squared minus a number x
5 squared is written as 5^2
Minus a number x means we subtract the variable x
[B]5^2 - x[/B]

5 times g reduced by the square of h

5 times g reduced by the square of h
Take this algebraic expression in pieces:
[LIST=1]
[*]5 times g means we multiply g by 5: 5g
[*]The square of h means we raise h to the 2nd power: h^2
[*]5 times g reduced by the square of h means we subtract h^2 from 5g:
[/LIST]
[B]5g - h^2[/B]

5×5 squared

5×5 squared
Determine index form
5^2 <-- index form
Evaluate:
5^2 = 5 * 5 = 25

6 is divided by square of a number

6 is divided by square of a number
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
the square of this means we raise x to the power of 2:
x^2
Next, we divide 6 by x^2:
[B]6/x^2[/B]

6 times j squared minus twice j squared

6 times j squared minus twice j squared
j squared means we raise the variable j to the power of 2:
j^2
6 times j squared means we multiply j^2 by 6:
6j^2
Twice j squared means we multiply j^2 by 2:
2j^2
The word [I]minus[/I] means we subtract 2j^2 from 6j^2
6j^2 - 2j^2
So if you must simplify, we group like terms and get:
(6 - 2)j^2
[B]4j^2[/B]

6 times y divided by x squared

6 times y divided by x squared
6 times y:
6y
x squared means we raise x to the power of 2:
x^2
The phrase [I]divided by[/I] means we have a fraction:
[B]6y/x^2[/B]

6 times y divided by x squared

6 times y divided by x squared
6 times y:
6y
x squared means we raise x to the power of 2:
x^2
The phrase [I]divided by[/I] means we divide 6y by x^2:
[B]6y/x^2[/B]

8 is subtracted from the square of x

8 is subtracted from the square of x
Take this algebraic expression in parts:
[LIST]
[*]The square of x means we raise x to the power of 2: x^2
[*]8 subtracted from the square of x is found by subtracting 8 from x^2
[/LIST]
[B]x^2 - 8[/B]

8 times 4 plus m squared

8 times 4 plus m squared
m squared means we raise m to the power of 2
m^2
4 plus m squared:
4 + m^2
8 times 4 plus m squared
[B]8(4 + m^2)[/B]

9 times x squared times y times z

9 times x squared times y times z
x squared:
x^2
x squared times y times z
x^2yz
9 times x squared times y times z
9x^2yz

A bag of fertilizer covers 300 square feet of lawn. Find how many bags of fertilizer should be purch

A bag of fertilizer covers 300 square feet of lawn. Find how many bags of fertilizer should be purchased to cover a rectangular lawn 290 feet by 150 feet.
The area of a rectangle is length * width, so we have:
A = 290 * 150
A = 43,500 sq ft.
Now, to find the number of bags needed for a 300 square feet per bag of fertilizer, we have:
Bags Needed = Total Square Feet of Lawn / Square Feet covered per bag
Bags Needed = 43,500 / 300
Bags Needed = [B]145[/B]

A bird was sitting 12 meters from the base of an oak tree and flew 15 meters to reach the top of the

A bird was sitting 12 meters from the base of an oak tree and flew 15 meters to reach the top of the tree. How tall is the tree?
So we have a [U]right triangle[/U]. Hypotenuse is 15. Base is 12. We want the length of the leg.
The formula for a right triangle relation of sides is a^2 + b^2 = c^2 where c is the hypotenuse and a, b are the sides
Rearranging this equation to isolate a, we get a^2 = c^2 - b^2
Taking the square root of both sides, we get a = sqrt(c^2 - b^2)
a = sqrt(15^2 - 12^2)
a = sqrt(225 - 144)
a = sqrt(81)
a = [B]9 meters[/B]

A carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional charge

A carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional charge of 25¢ per square foot for any footage that exceeds 180 sq ft and $1.30 per step for any carpeting on a staircase. A customers cleaning bill was $253.95. This included the cleaning of a staircase with 14 steps. In addition to the staircase, how many square feet of carpet did the customer have cleaned?
Calculate the cost of the staircase cleaning.
Staircase cost = $1.30 * steps
Staircase cost = $1.30 * 14
Staircase cost = $18.20
Subtract this from the cost of the total cleaning bill of $253.95. We do this to isolate the cost of the carpet.
Carpet cost = $253.95 - $18.20
Carpet cost = $235.75
Now, the remaining carpet cost can be written as:
75 + $0.25(s - 180) = $235.75 <-- were s is the total square foot of carpet cleaned
Multiply through and simplify:
75 + 0.25s - 45 = $235.75
Combine like terms:
0.25s + 30 = 235.75
[URL='https://www.mathcelebrity.com/1unk.php?num=0.25s%2B30%3D235.75&pl=Solve']Type this equation into our search engine[/URL] to solve for s, and we get:
s = [B]823[/B]

A cereal box has dimensions of 12" x 3" x 18". How many square inches of cardboard are used in its c

A cereal box has dimensions of 12" x 3" x 18". How many square inches of cardboard are used in its construction?
A cereal box is a rectangular solid. The volume formula is V = lwh.
Substituting these values of the cereal box in, we have:
V = 12(3)(18)
V = [B]648 cubic inches[/B]

A dog on a 20-foot long leash is tied to the middle of a fence that is 100 feet long. The dog ruined

A dog on a 20-foot long leash is tied to the middle of a fence that is 100 feet long. The dog ruined the grass wherever it could reach. What is the area of the grass that the dog ruined.
The leash forms a circle where the dog can get to.
A = pi(r)^2
A = 3.1415(20)^2
A = 3.1415 * 400
A = 1256 square feet
The fence blocks off half the circle where the dog can move to, so we have a half-circle area:
A = 1256/2
A = [B]628 square feet[/B]

A family room measures 15.6 feet long and 18.4 feet wide. What is the area of the room?

A family room measures 15.6 feet long and 18.4 feet wide. What is the area of the room?
The room is rectangular. So our area A = lw.
Using our [URL='https://www.mathcelebrity.com/rectangle.php?l=15.6&w=18.4&a=&p=&pl=Calculate+Rectangle']rectangle calculator[/URL], we get:
A = [B]287.04 square feet[/B]

A kitchen measures 5 yd by 6 yd. How much would it cost to install new linoleum in the kitchen if th

A kitchen measures 5 yd by 6 yd. How much would it cost to install new linoleum in the kitchen if the linoleum costs $2 per square foot?
The kitchen has an area of 5yd x 6yd = 30 sq yards.
If the linoleum costs $2 per square foot, we have 30 sq yards / $2 per square foot = [B]$15[/B]

A postcard is 4 inches tall and 5 inches wide. What is its area?

A postcard is 4 inches tall and 5 inches wide. What is its area?
A postcard is a rectangle. The area is 4 x 5 = [B]20 square inches[/B]

A rectangle is cut in half to create two squares that each has an area of 25. What is the perimeter

[SIZE=4]A rectangle is cut in half to create two squares that each has an area of 25. What is the perimeter of the original rectangle?
A. 20
B. 25
C. 30
D. 50[/SIZE]
[SIZE=4]Area of a square:
A = s^2
We're given A = 25:
s^2= 25
s = 5
This means the rectangle width is 5.
The rectangle length is 2(5) = 10.
Perimeter of a rectangle:
P = 2l + 2w
P = 2(10) + 2(5)
P = 20 + 10
P = [B]30 (choice C)[/B]
[B][/B]
[B][MEDIA=youtube]tKpS1gQY68o[/MEDIA][/B][/SIZE]

A rectangular hotel room is 4 yards by 5 yards. The owner of the hotel wants to recarpet the room wi

A rectangular hotel room is 4 yards by 5 yards. The owner of the hotel wants to recarpet the room with carpet that costs $76.00 per square yard. How much will it cost to recarpet the room? $
The area of a rectangle is length * width, so we have:
A = 5 yards * 4 yards
A = 20 square yards
Total cost = Cost per square yard * total square yards
Total Cost = $76 * 20
Total Cost = [B]$1520[/B]

A rectangular prism has a width of x feet, a length of y feet, and a height of h feet. Express its v

A rectangular prism has a width of [I]x[/I] feet, a length of [I]y[/I] feet, and a height of [I]h[/I] feet. Express its volume in square inches.
V = width * length * height
V = xyh
12 inches to a foot, so:
In cubic feet, we have 12 * 12 * 12 = 1728 cubic inches
V [B]= 1728xyh[/B]

A school wants to buy a chalkboard that measures 1 yard by 2 yards. The chalkboard costs $27.31 per

A school wants to buy a chalkboard that measures 1 yard by 2 yards. The chalkboard costs $27.31 per square yard. How much will the chalkboard cost?
Area of a chalkboard is denoted as :
A = lw
Given 1 yard width and 2 years length of the chalkboard, we have:
A = 2(1)
A = 2 square yards
Therefore, total cost is:
Total Cost = $27.31 * square yards
Total Cost = $27.31(2)
Total Cost = [B]$54.62[/B]

A square has a perimeter of 24 inches. What is the area of the square?

A square has a perimeter of 24 inches. What is the area of the square?
Perimeter of a square = 4s where s = the length of a side. Therefore, we have:
4s = P
4s = 24
Using our equation solver, [URL='https://www.mathcelebrity.com/1unk.php?num=4s%3D24&pl=Solve']we type in 4s = 24[/URL] and get:
s = 6
The problems asks for area of a square. It's given by
A = s^2
Plugging in s = 6, we get:
A = 6^2
A = 6 * 6
A = [B]36
[/B]
Now if you want a shortcut in the future, type in the shape and measurement you know. Such as:
[I][URL='https://www.mathcelebrity.com/square.php?num=24&pl=Perimeter&type=perimeter&show_All=1']square perimeter = 24[/URL][/I]
From the link, you'll learn every other measurement about the square.

A square of an integer is the integer. Find the integer.

A square of an integer is the integer. Find the integer.
Let the integer be n. The square means we raise n to the power of 2, so we have:
n^2 = n
Subtract n from each side:
n^2 - n = n - n
n^2 - n = 0
Factoring this, we get:
n(n - 1) = 0
So n is either [B]0 or 1[/B].

A times r squared multiplied by h

A times r squared multiplied by h
r squared means we raise r to the power of 2:
r^2
a times r squared:
ar^2
Multiplied by h:
[B]ahr^2[/B]

A triangle has an area of 60 square inches and a base of 10 inches. What is its height?

A triangle has an area of 60 square inches and a base of 10 inches. What is its height?
A = bh/2
b = 2A/h
b = 2(60)/10
b = 120/10
b = [B]12 inches[/B]

A triangular garden has base of 6 meters amd height of 8 meters. Find its area

A triangular garden has base of 6 meters amd height of 8 meters. Find its area
Area (A) of a triangle is:
A = bh/2
Plugging in our numbers, we get:
A = 6*8/2
A = [B]24 square meters[/B]

All squares are rectangles and all rectangles are parallelograms, therefore all squares are parallel

All squares are rectangles and all rectangles are parallelograms, therefore all squares are parallelograms. Is this true?
[B]Yes.[/B]
This is similar to A implies B and B implies C so A implies C also known as transitive property

Anna painted 1/6 of a wall, Eric painted 1/5 of the wall, and Meadow painted 1/4 of the wall. There

Anna painted 1/6 of a wall, Eric painted 1/5 of the wall, and Meadow painted 1/4 of the wall. There are now 3910 square feet left to paint. How many square feet did Anna paint?
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=4&num2=5&num3=6&pl=LCM']Using 60 as a common denominator through least common multiple[/URL], we get:
1/6 = 10/60
1/5 = 12/60
1/4 = 15/60
10/60 + 12/60 + 15/60 = 37/60
Remaining part of the wall is 60/60 - 37[B]/[/B]60 = 23/60
3910/23 = 170 for each 1/60 of a wall
Anna painted 1/6 or 10/60 of the wall. So we multiply 170 * 10 = [B]1,700 square feet[/B]

Approximate Square Root Using Exponential Identity

Free Approximate Square Root Using Exponential Identity Calculator - Calculates the square root of a positive integer using the Exponential Identity Method

Area Conversions

Free Area Conversions Calculator - This calculator converts between the following area measurements:

acre

hectare

square inch

square foot

square yard

square mile

square millimeter

square meter

square kilometer

acre

hectare

square inch

square foot

square yard

square mile

square millimeter

square meter

square kilometer

a^2 + b62 = c^2 for c

a^2 + b^2 = c^2 for c
Take the square root of each side:
c = [B]sqrt(a^2 + b^2)[/B]

Babylonian Method

Free Babylonian Method Calculator - Determines the square root of a number using the Babylonian Method.

Bakshali Method

Free Bakshali Method Calculator - Calculates the square root of a positive integer using the Bakshali Method

Basic Statistics

Free Basic Statistics Calculator - Given a number set, and an optional probability set, this calculates the following statistical items:

Expected Value

Mean = μ

Variance = σ^{2}

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Expected Value

Mean = μ

Variance = σ

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

c varies jointly as the square of q and cube of p

c varies jointly as the square of q and cube of p
The square of q means we raise q to the 2nd power:
q^2
The cube of p means we raise p to the rdd power:
p^3
The phrase [I]varies jointly[/I] means there exists a constant k such that:
[B]c = kp^3q^2[/B]

Chi-Square χ

Free Chi-Square χ^{2} Test Calculator - This calculator determines a χ^{2} chi-square test on a test statistic and determines if it is outside an accepted range with critical value test and conclusion.

Chi-Square Critical Values

Free Chi-Square Critical Values Calculator - Given a probability, this calculates the critical value for the right-tailed and left-tailed tests for the Chi-Square Distribution. CHIINV from Excel is used as well.

Complex Number Operations

Free Complex Number Operations Calculator - Given two numbers in complex number notation, this calculator:

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit

Free Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit Calculator - Given two distributions X and Y, this calculates the following:

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r^{2}

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

d squared is greater than or equal to 17

d squared is greater than or equal to 17
d squared means we raise the variable d to the power of 2:
d^2
The phrase [I]greater than or equal to[/I] means an inequality. So we set this up using the >= in relation to 17:
[B]d^2 >= 17[/B]

difference between 2 positive numbers is 3 and the sum of their squares is 117

difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant Δ:[/U]
Δ = b2 - 4ac:
Δ = 62 - 4 x 2 x -108
Δ = 36 - -864
Δ = 900 <--- Discriminant
Since Δ is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant Δ:[/U]
√Δ = √(900)
√Δ = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + √Δ
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - √Δ
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Difference of Two Squares

Free Difference of Two Squares Calculator - Factors a difference of squares binomial in the form a^{2} - b^{2} or multiplies 2 binomials through in the form (ax + by)(ax - by).

Divide the sum of a and b by the square of c

Divide the sum of a and b by the square of c
The sum of a and b:
a + b
The square of c means we raise c to the power of 2:
c^2
Divide means we have a quotient, with a + b on top, and c^2 on the bottom:
[B](a + b)/c^2[/B]

divide the sum of the square of a and b by thrice c

divide the sum of the square of a and b by thrice c
Sum of the squares of a and b is found as follows:
[LIST]
[*]a squared means we raise a to the power of 2: a^2
[*]b squared means we raise b to the power of 2: b^2
[*]Sum of the squares means we add both terms: a^2 + b^2
[*]Thrice c means we multiply c by 3: 3c
[/LIST]
Divide means we have a quotient:
[B](a^2 + b^2)/3c[/B]

Divide the sum of the squares of a and b by the square of c

Divide the sum of the squares of a and b by the square of c
square of a:
a^2
square of b:
b^2
Sum of the squares of a and b:
a^2 + b^2
square of c:
c^2
Divide the Sum of the squares of a and b by the square of c:
[B](a^2 + b^2)/c^2[/B]

Each side of a square is lengthened by 3 inches . The area of this new, larger square is 25 square

Each side of a square is lengthened by 3 inches . The area of this new, larger square is 25 square inches. Find the length of a side of the original square.
area of a square is s^2
New square has sides s + 3, so the area of 25 is:
(s + 3)^2 = 25
[URL='https://www.mathcelebrity.com/1unk.php?num=%28s%2B3%29%5E2%3D25&pl=Solve']Solving for s[/URL], we get:
s = [B]2[/B]

Equation and Inequalities

Free Equation and Inequalities Calculator - Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

Estimate Square Roots

Free Estimate Square Roots Calculator - Estimates the square root of a number

Explain the relationship between "squaring" a number and finding the "square root" of a number. Use

Explain the relationship between "squaring" a number and finding the "square root" of a number. Use an example to further explain your answer.
Squaring a number means raising it to the power of 2
The square root of a number [I]undoes[/I] a square of a number.
So square root of x^2 is x
x squared is x^2
Let x = 5.
x squared = 5^2 = 25
Square root of 25 = square root of 5^2 = 5

f varies jointly with u and h and inversely with the square of y.

f varies jointly with u and h and inversely with the square of y.
Variation means we have a constant k.
Varies jointly with u and h means we multiply k by hu
Varies inversely with the square of y means we divide by y^2
[B]f = khu/y^2[/B]

Factoring and Root Finding

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential*positive* and *negative* roots using Descarte’s Rule of Signs

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential

Fantasia decided to paint her circular room which had a diameter of 25 feet. She started painting in

Fantasia decided to paint her circular room which had a diameter of 25 feet. She started painting in the center and when she had painted a circle with a 5-foot diameter, she used one quart of paint. How many more quarts of paint must Fantasia buy to finish her room?
The area formula for a circle is:
Area = pir^2
Area of full room
Radius = D/2
Radius = 25/2
Radius = 12.5
Area = 3.1415 * 12.5 * 12.5
Area = 490.625
Area of 5-foot diameter circle
Radius = D/2
Radius = 5/2
Radius = 2.5
Area = 3.1415 * 2.5 * 2.5
Area = 19.625
So 1 quart of paint covers 19.625 square feet
Area of unpainted room = Area of Room - Area of 5-foot diameter circle
Area of unpainted room = 490.625 - 19.625
Area of unpainted room = 471
Calculate quarts of paint needed:
Quarts of paint needed = Area of unpainted Room / square feet per quart of paint
Quarts of paint needed = 471/19.625
Quarts of paint needed = [B]24 quarts[/B]

Find Mean 106 and standard deviation 10 of the sample mean which is 25

Do you mean x bar?
mean of 106 inches and a standard deviation of 10 inches and for sample of size is 25. Determine the mean and the standard deviation of /x
If so, x bar equals the population mean. So it's [B]106[/B].
Sample standard deviation = Population standard deviation / square root of n
10/Sqrt(25)
10/5
[B]2[/B]

find the two square roots of 81

find the two square roots of 81
When we multiply 9 * 9, we get 81
When we multiply -9 * -9, we get 81
So our two square roots of 81 are:
[LIST]
[*][B]-9, 9[/B]
[/LIST]

Find two consecutive integers if the sum of their squares is 1513

Find two consecutive integers if the sum of their squares is 1513
Let the first integer be n. The next consecutive integer is (n + 1).
The sum of their squares is:
n^2 + (n + 1)^2 = 1513
n^2 + n^2 + 2n + 1 = 1513
2n^2 + 2n + 1 = 1513
Subtract 1513 from each side:
2n^2 + 2n - 1512 = 0
We have a quadratic equation. We [URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B2n-1512%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this into our search engine[/URL] and get:
n = (-27, 28)
Let's take the positive solution.
The second integer is: n + 1
28 + 1 = 29

Find two consecutive odd integers such that the sum of their squares is 290

Find two consecutive odd integers such that the sum of their squares is 290.
Let the first odd integer be n.
The next odd integer is n + 2
Square them both:
n^2
(n + 2)^2 = n^2 + 4n + 4 from our [URL='https://www.mathcelebrity.com/expand.php?term1=%28n%2B2%29%5E2&pl=Expand']expansion calculator[/URL]
The sum of the squares equals 290
n^2 + n^2 + 4n + 4 = 290
Group like terms:
2n^2 + 4n + 4 = 290
[URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B4n%2B4%3D290&pl=Solve+Quadratic+Equation&hintnum=+0']Enter this quadratic into our search engine[/URL], and we get:
n = 11, n = -13
Which means the two consecutive odd integer are:
11 and 11 + 2 = 13. [B](11, 13)[/B]
-13 and -13 + 2 = -11 [B](-13, -11)[/B]

Find two consecutive positive integers such that the difference of their square is 25

Find two consecutive positive integers such that the difference of their square is 25.
Let the first integer be n. This means the next integer is (n + 1).
Square n: n^2
Square the next consecutive integer: (n + 1)^2 = n^2 + 2n + 1
Now, we take the difference of their squares and set it equal to 25:
(n^2 + 2n + 1) - n^2 = 25
Cancelling the n^2, we get:
2n + 1 = 25
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B1%3D25&pl=Solve']Typing this equation into our search engine[/URL], we get:
n = [B]12[/B]

Find two consecutive positive integers such that the sum of their squares is 25

Find two consecutive positive integers such that the sum of their squares is 25.
Let the first integer be x. The next consecutive positive integer is x + 1.
The sum of their squares equals 25. We write this as::
x^2 + (x + 1)^2
Expanding, we get:
x^2 + x^2 + 2x + 1 = 25
Group like terms:
2x^2 + 2x + 1 = 25
Subtract 25 from each side:
2x^2 + 2x - 24 = 0
Simplify by dividing each side by 2:
x^2 + x - 12 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get x = 3 or x = -4. The problem asks for positive integers, so we discard -4, and use 3.
This means, our next positive integer is 3 + 1 = 4. So we have [B](3, 4) [/B]as our answers.
Let's check our work:
3^2 + 4^2 = 9 + 16 = 25

Football Squares

Free Football Squares Calculator - Generates a Football Squares grid

Half of g multiplied by t squared is equal to d.

Half of g multiplied by t squared is equal to d.
Half of g:
g/2
t squared:
t^2
Half of g multiplied by t squared:
gt^2/2
The phrase [I]is equal to[/I] mean we set gt^2/2 equal to d:
[B]gt^2/2 = d[/B]

Hardy-Weinberg

Free Hardy-Weinberg Calculator - Given a dominant gene frequency probability of p, this displays the Punnet Square Hardy Weinberg frequencies

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of columns. Find the number of rows and columns.
Let r be the number of rows and c be the number of columns. We have the area:
rc = 324
Since rows equal columns, we have a square, and we can set r = c.
c^2 = 324
Take the square root of each side:
[B]c = 18[/B]
Which means [B]r = 18[/B] as well.
What we have is a garden of 18 x 18.

Help on problem

[B]I need 36 m of fencing for my rectangular garden. I plan to build a 2m tall fence around the garden. The width of the garden is 6 m shorter than twice the length of the garden. How many square meters of space do I have in this garden?
List the answer being sought (words) ______Need_________________________
What is this answer related to the rectangle?_Have_________________________
List one piece of extraneous information____Need_________________________
List two formulas that will be needed_______Have_________________________
Write the equation for width_____________Have_________________________
Write the equation needed to solve this problem____Need____________________[/B]

How many inches are in a square foot?

How many inches are in a square foot?
Square area is s * s where s = 12 inches per foot
Area = 12 inches per foot * 12 inches per foot
Area = [B]144 square inches[/B]

How many square feet are in a square yard?

How many square feet are in a square yard?
A square yard = 1 yard * 1 yard
Since 1 yard = 3 feet, we have:
A square yard = 3 feet * 3 feet = [B]9 feet[/B]

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integ

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integer.
[LIST]
[*]Let the integer be "x".
[*]Square the integer: x^2
[*]7 times the square: 7x^2
[*]5 times the integer: 5x
[*]Add them together: 7x^2 + 5x
[*][I]The result is[/I] means an equation, so we set 7x^2 + 5x equal to 2
[/LIST]
7x^2 + 5x = 2
[U]This is a quadratic equation. To get it into standard form, we subtract 2 from each side:[/U]
7x^2 + 5x - 2 = 2 - 2
7x^2 + 5x - 2 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=7x%5E2%2B5x-2%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get two solutions:
[LIST=1]
[*]x = 2/7
[*]x= -1
[/LIST]
The problem asks for an integer, so our answer is x[B] = -1[/B].
[U]Let's check our work by plugging x = -1 into the quadratic:[/U]
7x^2 + 5x - 2 = 0
7(-1)^2 + 5(-1) - 2 ? 0
7(1) - 5 - 2 ? 0
0 = 0
So we verified our answer, [B]x = -1[/B].

If 800 feet of fencing is available, find the maximum area that can be enclosed.

If 800 feet of fencing is available, find the maximum area that can be enclosed.
Perimeter of a rectangle is:
2l + 2w = P
However, we're given one side (length) is bordered by the river and the fence length is 800, so we have:
So we have l + 2w = 800
Rearranging in terms of l, we have:
l = 800 - 2w
The Area of a rectangle is:
A = lw
Plug in the value for l in the perimeter into this:
A = (800 - 2w)w
A = 800w - 2w^2
Take the [URL='https://www.mathcelebrity.com/dfii.php?term1=800w+-+2w%5E2&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']first derivative[/URL]:
A' = 800 - 4w
Now set this equal to 0 for maximum points:
4w = 800
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%3D800&pl=Solve']Typing this equation into the search engine[/URL], we get:
w = 200
Now plug this into our perimeter equation:
l = 800 - 2(200)
l = 800 - 400
l = 400
The maximum area to be enclosed is;
A = lw
A = 400(200)
A = [B]80,000 square feet[/B]

if a number is added to its square, it equals 20

if a number is added to its square, it equals 20.
Let the number be an arbitrary variable, let's call it n.
The square of the number means we raise n to the power of 2:
n^2
We add n^2 to n:
n^2 + n
It equals 20 so we set n^2 + n equal to 20
n^2 + n = 20
This is a quadratic equation. So [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn%3D20&pl=Solve+Quadratic+Equation&hintnum=+0']we type this equation into our search engine[/URL] to solve for n and we get two solutions:
[B]n = (-5, 4)[/B]

if a number is added to its square, the result is 72. find the number

if a number is added to its square, the result is 72. find the number.
Let the number be n. We're given:
n + n^2 = 72
Subtract 72 from each side, we get:
n^2 + n - 72 = 0
This is a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this equation into our search engine[/URL], and we get:
[B]n = 8 and n = -9[/B]

If all A's are B's, then all B's are A's. Is this true?

If all A's are B's, then all B's are A's. Is this true?
[B]No.[/B]
Example:
All dogs are mammals, but not all mammals are dogs.
All squares are rectangles, but not all rectangles are squares.

if i = square root of -1 what is the sum (7 + 3i) + (-8 + 9i)

if i = square root of -1 what is the sum (7 + 3i) + (-8 + 9i)
We group like terms, and we get:
7 - 8 + (3 + 9)i
Simplifying, we get:
[B]-1 + 12i[/B]

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equ

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equal to 2.
We set up the variation equation with a constant k such that:
p = k/q^2 [I](inversely proportional means we divide)
[/I]
When q is 4 and p is 2, we have:
2 = k/4^2
2 = k/16
Cross multiply:
k = 2 * 16
k = 32
Now, the problem asks for p when q = 2:
p = 32/2^2
p = 32/4
p = [B]8
[MEDIA=youtube]Mro0j-LxUGE[/MEDIA][/B]

If the perimeter of square C is triple the perimeter of square D, the area of square C is how many t

If the perimeter of square C is triple the perimeter of square D, the area of square C is how many times the area of square D?
A. 1/3
B. 1
C. 3
D. 9
E. 27
Let side of D be s. Perimeter is 4s. Area is (4s)^2 = 16s^2
Let side of C be 3s. Perimeter is 4(3s) = 12s. Area is (12s)^2 = 144s^2
144s^2/16s^2 = [B]9 times bigger[/B]

if x^2=y^3, for what value of z does x^{3z}= y^9

if x^2=y^3, for what value of z does x^{3z}= y^9
y^9 = y^3 * y^3, so if we square the right side, we must square the left side for equivalence:
x^2 * x^2 = x^4
Therefore,
x^4 = y^9
Going back to our problem, x^{3z}= y^9, so 3z = 4
Divide each side by 3 to isolate z, and we have:
3z/3 = 4/3
z = [B]4/3[/B]

In a hurricane the wind pressure varies directly as the square of the wind velocity. If a wind pres

In a hurricane the wind pressure varies directly as the square of the wind velocity. If a wind pressure is a measure of a hurricane's destruction capacity, what happens to this destructive power when the wind speed doubles?
Let P = pressure and v = velocity (wind speed)
We are given p = v^2
Double velocity, so we have a new pressure P2:
P2 = (2v)^2
P2 = 4v^2
Compare the 2:
p = v^2
p = 4v^2
Doubling the wind speed [B]quadruples, or 4 times[/B] the pressure.

It took 3.5 gallons of paint to cover a wall that is 985 square feet. How many gallons will it take

It took 3.5 gallons of paint to cover a wall that is 985 square feet. How many gallons will it take to cover a wall that is 6501 square feet?
Set up a proportion of gallons of paint to square feet where n is the number of gallons of paint to cover 6501 square feet
3.5/985 = n/6501
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=3.5&num2=n&den1=985&den2=6501&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get:
n = [B]23.1[/B]

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered fro

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered from 1 to 25. What is the probability of drawing a number that is square?
The squares from 1 - 25 less than or equal to 25 are as follows:
[LIST=1]
[*]1^2 = 1
[*]2^2 = 4
[*]3^2 = 9
[*]4^2 = 16
[*]5^2 = 25
[/LIST]
So the following 5 cards are squares:
{1, 4, 9, 16, 25}
Therefore, our probability of drawing a square is:
P(square) = Number of Squares / Number of Cards
P(square) = 5/25
This fraction can be simplified. So [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F25&frac2=3%2F8&pl=Simplify']we type in 5/25 into our search engine, choose simplify[/URL], and we get:
P(square) = [B]1/5[/B]

Juan runs out of gas in a city. He walks 30yards west and then 16 yards south looking for a gas stat

Juan runs out of gas in a city. He walks 30yards west and then 16 yards south looking for a gas station. How far is he from his starting point?
Juan is located on a right triangle. We calculate the hypotenuse:
30^2 + 16^2 = Hypotenuse^2
900 + 256 = Hypotenuse^2
Hypotenuse^2 = 1156
Take the square root of each side:
[B]Hypotenuse = 34 yards[/B]

K varies inversely with square root of m and directly with the cube of n.

K varies inversely with square root of m and directly with the cube of n.
[LIST]
[*]We take a constant c as our constant of proportionality.
[*]The word inversely means we divide
[*]The word directly means we multiply
[/LIST]
[B]k = cn^3/sqrt(m)[/B]

Kamara has a square fence kennel area for her dogs in the backyard. The area of the kennel is 64 ft

Kamara has a square fence kennel area for her dogs in the backyard. The area of the kennel is 64 ft squared. What are the dimensions of the kennel? How many feet of fencing did she use? Explain.
Area of a square with side length (s) is:
A = s^2
Given A = 64, we have:
s^2 = 64
[URL='https://www.mathcelebrity.com/radex.php?num=sqrt(64%2F1)&pl=Simplify+Radical+Expression']Typing this equation into our math engine[/URL], we get:
s = 8
Which means the dimensions of the kennel are [B]8 x 8[/B].
How much fencing she used means perimeter. The perimeter P of a square with side length s is:
P = 4s
[URL='https://www.mathcelebrity.com/square.php?num=8&pl=Side&type=side&show_All=1']Given s = 8, we have[/URL]:
P = 4 * 8
P = [B]32[/B]

Kamille is calculating the length of diagonal on a picture board and gets a solution of the square r

Kamille is calculating the length of diagonal on a picture board and gets a solution of the square root of 58. She needs to buy the ribbon to put across the diagonal of the board, so she estimates that she will need at least 60 inches of ribbon to cover the diagonal. Is she correct? Explain.
[URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%2858%29&pl=Calculate']The square root of 58 [/URL]has an answer between 7 and 8.
So Kamille is [B]incorrect[/B]. She needs much less than 60 inches of ribbon. She needs less than 8 inches of ribbon.

Keith is cutting two circular table tops out of a piece of plywood. the plywood is 4 feet by 8 feet

Keith is cutting two circular table tops out of a piece of plywood. the plywood is 4 feet by 8 feet and each table top has a diameter of 4 feet. If the price of a piece of plywood is $40, what is the value of the plywood that is wasted after the table tops are cut?
Area of the plywood = 4 * 8 = 32 square feet
[U]Calculate area of 1 round top[/U]
Diameter = 2
Radius = Diameter/2 = 4/2 = 2
Area of each round top = pir^2
Area of each round top = 3.14 * 2 * 2
Area of each round top = 12.56 square feet
[U]Calculate area of 2 round tops[/U]
Area of 2 round tops = 12.56 + 12.56
Area of 2 round tops = 25.12 sq feet
[U]Calculate wasted area:[/U]
Wasted area = area of the plywood - area of 2 round tops
Wasted area = 32 - 25.12
Wasted area = 6.88 sq feet
[U]Calculate cost per square foot of plywood:[/U]
Cost per sq foot of plywood = Price per plywood / area of the plywood
Cost per sq foot of plywood = 40/32
Cost per sq foot of plywood = $1.25
[U]Calculate the value of the plywood:[/U]
Value of the plywood = Wasted Area sq foot * Cost per sq foot of plywood
Value of the plywood = 6.88 * 1.25
Value of the plywood = [B]$8.60[/B]

Kris wants to fence in her square garden that is 40 feet on each side. If she places posts every 10

Kris wants to fence in her square garden that is 40 feet on each side. If she places posts every 10 feet, how many posts will she need?
Perimeter (P) of a square with side s:
P = 4s
Given s = 40, we have:
P = 4(40)
P = 160 feet
160 feet / 10 foot spaces = [B]16 posts[/B]

Lagrange Four Square Theorem (Bachet Conjecture)

Free Lagrange Four Square Theorem (Bachet Conjecture) Calculator - Builds the Lagrange Theorem Notation (Bachet Conjecture) for any natural number using the Sum of four squares.

Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden

Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden or a round garden with the fencing.
Laura did some calculations and found that a circular garden would give her 1380 more square feet than a square garden. How many feet of fencing were in the roll that Laura found? (Round to the nearest foot.)
Feet of fencing = n
Perimeter of square garden = n
Each side of square = n/4
Square garden's area = (n/4)^2 = n^2/16
Area of circle garden with circumference = n is:
Circumference = pi * d
n = pi * d
Divide body tissues by pi:
d = n/pi
Radius = n/2pi
Area = pi * n/2pi * n/2pi
Area = pin^2/4pi^2
Reduce by canceling pi:
n^2/4pi
n^2/4 * 3.14
n^2/12.56
The problem says that the difference between the square's area and the circle's area is equal to 1380 square feet.
Area of Circle - Area of Square = 1380
n^2/12.56 - n^2/16 = 1380
Common denominator = 200.96
(16n^2 - 12.56n^2)/200.96 = 1380
3.44n^2/200.96 = 1380
Cross multiply:
3.44n^2 = 277,324.8
n^2 = 80,617.7
n = 283.9
Nearest foot = [B]284[/B]

Let n be an integer. If n^2 is odd, then n is odd

Let n be an integer. If n^2 is odd, then n is odd
Proof by contraposition:
Suppose that n is even. Then we can write n = 2k
n^2 = (2k)^2 = 4k^2 = 2(2k) so it is even
[I]So an odd number can't be the square of an even number. So if an odd number is a square it must be the square of an odd number.[/I]

m is inversely proportional to the square of p-1 when p=4 m=5 find m when p=6

m is inversely proportional to the square of p-1 when p=4 and m=5. find m when p=6
Inversely proportional means there is a constant k such that:
m = k/(p - 1)^2
When p = 4 and m = 5, we have:
5 = k/(4 - 1)^2
5 = k/3^2
5 = k/9
[U]Cross multiply:[/U]
k = 45
[U]The problems asks for m when p = 6. And we also now know that k = 45. So plug in the numbers:[/U]
m = k/(p - 1)^2
m = 45/(6 - 1)^2
m = 45/5^2
m = 45/25
m = [B]1.8[/B]

Multiply c by five and square the answer

Multiply c by five and square the answer
Multiply c by five:
5c
Square the answer means we raise 5c to the power of 2:
[B](5c)^2 [/B]
This can also be written as [B]25c^2[/B]

N squared multiplied by the difference of n and 3

N squared multiplied by the difference of n and 3
n squared means we raise n to the power of 2:
n^2
The difference of n and 3 means we subtract 3 from n:
n - 3
Now we multiply both terms together:
[B]n^2(n - 3)[/B]

Newton Method

Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method

Nine workers are hired to harvest potatoes from a field. Each is given a plot which is 5x5 feet in s

Nine workers are hired to harvest potatoes from a field. Each is given a plot which is 5x5 feet in size. What is the total area of the field?
Area of each plot is 5x5 = 25 square feet.
Total area = Area per plot * number of plots
Total area = 25 sq ft * 9
Total area = [B]225 sq ft[/B]

Number Property

Free Number Property Calculator - 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

n^2 + 9 = 34

n^2 + 9 = 34
Subtract 9 from each side:
n^2 + 9 - 9 = 34 - 9
n^2 = 25
Take the square root of each side:
n = [B]5[/B]

n^2 - 1 = -99/100

n^2 - 1 = -99/100
Add 1 (100/100) to each side:
n^2 - 1 + 1 = -99/100 + 100/100
Cancel the 1's on the left side:
n^2 = 1/100
Take the square root of both sides:
n = [B]1/10 or -1/10[/B]

n^2 = 1/4

n^2 = 1/4
Take the square root of each side:
n = [B]1/2[/B]

n^2 = 6&1/4

n^2 = 6&1/4
[URL='https://www.mathcelebrity.com/fraction.php?frac1=6%261%2F4&frac2=3%2F8&pl=Simplify']6&1/4[/URL] = 25/4
n^2 = 25/4
Take the square root of each side:
n = [B]5/2 or -5/2[/B]

n^2 = 64

n^2 = 64
Take the square root of each side:
sqrt(n^2) = sqt(64)
n = [B]8[/B]

One fifth of the square of a number

One fifth of the square of a number
We have an algebraic expression. Let's break this into parts.
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The square of a number means we raise it to the power of 2. So we have x^2
[*]One-fifth means we have a fraction, where we divide our x^2 in Step 2 by 5. So we get our final answer below:
[/LIST]
[B]x^2/5[/B]

One number is equal to the square of another. Find the numbers if both are positive and their sum is

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650
Let the number be n. Then the square is n^2. We're given:
n^2 + n = 650
Subtract 650 from each side:
n^2 + n - 650 = 0
We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get:
n = 25 and n = -26
Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution.
the second solution is 25^2 = [B]625[/B]

p more than the square of q

p more than the square of q
Take this algebraic expression in parts:
Step 1: Square of q means raise q to the 2nd power:
q^2
Step 2: The phrase [I]more[/I] means we add p to q^2
[B]q^2 + p[/B]

P varies directly as q and the square of r and inversely as s

P varies directly as q and the square of r and inversely as s
There exists a constant k such that:
p = kqr^2/s
[I]Note: Direct variations multiply and inverse variations divide[/I]

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6, what is the equation of variation?
Two rules of variation:
[LIST=1]
[*]Varies directly means we multiply
[*]Varies inversely means we divide
[/LIST]
There exists a constant k such that our initial equation of variation is:
p = kr^2/qs
[B][/B]
With p = 40 when q = 5, r = 4 and s = 6, we have:
4^2k / 5 * 6 = 40
16k/30 = 40
Cross multiply:
16k = 40 * 30
16k = 1200
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=16k%3D1200&pl=Solve']equation calculator[/URL], we get:
k = [B]75[/B]
So our final equation of variation is:
[B]p = 75r^2/qs[/B]

Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job

Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
Let y be the number of square yards. We have the following equation:
0.2y + 10 = 25
To solve for y, we[URL='https://www.mathcelebrity.com/1unk.php?num=0.2y%2B10%3D25&pl=Solve'] type this equation into our search engine [/URL]and we get:
y = [B]75[/B]

Prove sqrt(2) is irrational

Use proof by contradiction. Assume sqrt(2) is rational.
This means that sqrt(2) = p/q for some integers p and q, with q <>0.
We assume p and q are in lowest terms.
Square both side and we get:
2 = p^2/q^2
p^2 = 2q^2
This means p^2 must be an even number which means p is also even since the square of an odd number is odd.
So we have p = 2k for some integer k. From this, it follows that:
2q^2 = p^2 = (2k)^2 = 4k^2
2q^2 = 4k^2
q^2 = 2k^2
q^2 is also even, therefore q must be even.
So both p and q are even.
This contradicts are assumption that p and q were in lowest terms.
So sqrt(2) [B]cannot be rational.
[MEDIA=youtube]tXoo9-8Ewq8[/MEDIA][/B]

Prove that the difference between alternate consecutive squares as always even

Take an integer n. The next alternate consecutive integer is n + 2
Subtract the difference of the squares:
(n + 2)^2 - n^2
n^2 + 4n + 4 - n^2
n^2 terms cancel, we get:
4n + 4
Factor out a 4:
4(n + 1)
If n is odd, n + 1 is even. 4 * even is always even
If n is even, n + 1 is odd. 4 * odd is always odd
Since both cases are even, we've proven our statement.
[MEDIA=youtube]J_E9lR5qFY0[/MEDIA]

Prove the difference between two consecutive square numbers is always odd

Take an integer n. The next consecutive integer is n + 1
Subtract the difference of the squares:
(n + 1)^2 - n^2
n^2 + 2n + 1 - n^2
n^2 terms cancel, we get:
2n + 1
2 is even. For n, if we use an even:
we have even * even = Even
Add 1 we have Odd
2 is even. For n, if we use an odd:
we have even * odd = Even
Add 1 we have Odd
Since both cases are odd, we've proven our statement.
[MEDIA=youtube]RAi0HbH5bqc[/MEDIA]

Punnett Square

Free Punnett Square Calculator - Completes a punnet square based on genotypes

Quadratic Equations and Inequalities

Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax^{2} + bx + c = 0. Also generates practice problems as well as hints for each problem.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)^{2} + k

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

r squared plus the product of 3 and s plus 5

r squared plus the product of 3 and s plus 5
r squared means we raise r to the power of 2
r^2
The product of 3 and s means we multiply s by 3:
3s
plus 5 means we add
3s + 5
R squared plus means we add r^2:
[B]r^2 + 3s + 5[/B]

r varies directly with s and inversely with the square root of t

r varies directly with s and inversely with the square root of t
Varies directly means we multiply
Varies inversely means we divide
There exists a constant k such that:
[B]r = ks/sqrt(t)[/B]

ratio of the squares of t and u

ratio of the squares of t and u
Ratio is also known as quotient in algebraic expression problems.
The square of t means we raise t to the power of 2:
t^2
The square of u means we raise u to the power of 2:
u^2
ratio of the squares of t and u means we divide t^2 by u^2:
[B]t^2/u^2[/B]

Rational,Irrational,Natural,Integer Property

Free Rational,Irrational,Natural,Integer Property Calculator - This calculator takes a number, decimal, or square root, and checks to see if it has any of the following properties:

* Integer Numbers

* Natural Numbers

* Rational Numbers

* Irrational Numbers Handles questions like: Irrational or rational numbers Rational or irrational numbers rational and irrational numbers Rational number test Irrational number test Integer Test Natural Number Test

* Integer Numbers

* Natural Numbers

* Rational Numbers

* Irrational Numbers Handles questions like: Irrational or rational numbers Rational or irrational numbers rational and irrational numbers Rational number test Irrational number test Integer Test Natural Number Test

rs+h^2=1 for h

rs+h^2=1 for h
Subtract rs from each side to isolate h:
rs - rs + h^2 = 1 - rs
Cancel the rs on the left side:
h^2 = 1 - rs
Take the square root of each side:
sqrt(h^2) = sqrt(1 - rs)
[B]h = +- sqrt(1 -rs)[/B]

s = tu^2 for u

s = tu^2 for u
Divide each side by t
u^2 = s/t
Take the square root of each side
[LIST]
[*]u = sqrt(s/t)
[*]u = -sqrt(s/t)
[/LIST]
We have two answers due to negative number squared is positive

Simplify 7sqrt(3) - sqrt(12)

7sqrt(3) is broken down.
sqrt(12) is not broken down. Let's find all the factors of 12 and see if we stumble on a perfect square:
[LIST]
[*]1 * 12
[*]2 * 6
[*]3 * 4
[/LIST]
4 is a perfect square, since sqrt(4) = 2.
So sqrt(12) = sqrt(3 * 4)
We pull the sqrt(4) = 2 outside the radical and rewrite our problem as:
7sqrt(3) - 2sqrt(3)
These are like terms, so we have:
(7 - 2)sqrt(3)
[B]5sqrt(3) [/B]
[MEDIA=youtube]ljXVXWnKiWY[/MEDIA]

Six is the principal square root of 36

Six is the principal square root of 36
The two square roots of 36 are:
[LIST]
[*]+6
[*]-6
[/LIST]
The positive square root is known as the principal square root, therefore, this is [B]true[/B].

Solve for h. rs + h^2 = l

Solve for h. rs + h^2 = l
[U]Subtract rs from each side to isolate h:[/U]
rs - rs + h^2 = l - rs
[U]Cancel the rs terms on the left side, and we get:[/U]
h^2 = l - rs
[U]Take the square root of each side:[/U]
h = [B]sqrt(l - rs)[/B]

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v
1/2(2/5) = 1/5 since the 2's cancel
r^2/r^2 = 1
So we simplify, and get:
mgh=1/2mv^2+1/5(mv^2) for v
Divide each side by m, so m's cancel in each term on the left and right side:
gh = 1/2v^2 + 1/5(v^2)
Combine like terms for v^2 on the right side:
1/2 + 1/5 = 7/10 from our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction calculator[/URL]
So we have:
gh = 7v^2/10
Multiply each side by 10:
10gh = 7v^2
Now divide each side by 7
10gh/7 = v^2
Take the square root of each side:
[B]v = sqrt(10gh/7)[/B]

Square Number

Free Square Number Calculator - This calculator determines the nth square number

Square root of 9136 divided by 43

Square root of 9136 divided by 43
First, [URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%289136%29&pl=Calculate']take the square root of 9136 in our calculator[/URL]:
4 * sqrt(571)
Now divide this by 43:
[B]4 * sqrt(571) / 43[/B]

square root of the sum of 2 variables

square root of the sum of 2 variables
The phrase [I]2 variables[/I] means we choose 2 arbitrary variables, let's call them x and y:
x, y
The sum of 2 variables means we add:
x + y
Square root of the sum of 2 variables is written as:
[B]sqrt(x + y)[/B]

square root of x times the square root of y

square root of x times the square root of y
square root of x:
sqrt(x)
square root of y:
sqrt(y)
square root of x times the square root of y
[B]sqrt(x) * sqrt(y)[/B]

Square Root Table

Free Square Root Table Calculator - Generates a square root table for the first (n) numbers rounded to (r) digits

Square Roots and Exponents

Free Square Roots and Exponents Calculator - Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x^{th} power denoted as n^{x} (Write without exponents)

* n raised to the x^{th} power raised to the yth power denoted as (n^{x})^{y} (Write without exponents)

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x

* n raised to the x

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

Squares

Free Squares Calculator - Solve for Area of a square, Perimeter of a square, side of a square, diagonal of a square.

standard deviation of 545 dollars. Find the sample size needed to have a confidence level of 95% and

Standard Error (margin of Error) = Standard Deviation / sqrt(n)
128 = 545/sqrt(n)
Cross multiply:
128sqrt(n) = 545
Divide by 128
sqrt(n) = 4.2578125
Square both sides:
[B]n = 18.1289672852 But we need an integer, so the answer is 19[/B]

Subtract 12 from the square sum of w and v

Sum of w and v:
w + v
Square that sum
(w + v)^2
Subtract 12 from the squared sum
(w + v)^2 - 12

Sum of the First (n) Numbers

Free Sum of the First (n) Numbers Calculator - Determines the sum of the first (n)

* Whole Numbers

* Natural Numbers

* Even Numbers

* Odd Numbers

* Square Numbers

* Cube Numbers

* Fourth Power Numbers

* Whole Numbers

* Natural Numbers

* Even Numbers

* Odd Numbers

* Square Numbers

* Cube Numbers

* Fourth Power Numbers

sum of the squares of u and v

sum of the squares of u and v
The square of u means we raise u to the power of 2
u^2
The square of v means we raise v to the power of 2
v^2
The sum means we add v^2 to u^2:
[B]u^2 + v^2[/B]

t varies directly with the square of r and inversely with w

t varies directly with the square of r and inversely with w
There exists a constant k such that:
[B]t = kr^2/w[/B]
[I]Directly means multiply and inversely means divide[/I]

Take a look at the following sums: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 +

Take a look at the following sums:
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
a. Come up with a conjecture about the sum when you add the first *n* odd numbers. For example, when you added the first 5 odd numbers (1 + 3 + 5 + 7 + 9), what did you get? What if wanted to add the first 10 odd numbers? Or 100?
b. Can you think of a geometric interpretation of this pattern? If you start with one square and add on three more, what can you make? If you now have 4 squares and add on 5 more, what can you make?
c. Is there a similar pattern for adding the first n even numbers?
2 = 2
2 + 4 = 6
2 + 4 + 6 = 12
2 + 4 + 6 + 8 = 20
a. The formula is [B]n^2[/B].
The sum of the first 10 odd numbers is [B]100[/B] seen on our s[URL='http://www.mathcelebrity.com/sumofthefirst.php?num=10&pl=Odd+Numbers']um of the first calculator[/URL]
The sum of the first 100 odd numbers is [B]10,000[/B] seen on our [URL='http://www.mathcelebrity.com/sumofthefirst.php?num=100&pl=Odd+Numbers']sum of the first calculator[/URL]
b. Geometric is 1, 4, 9 which is our [B]n^2[/B]
c. The sum of the first n even numbers is denoted as [B]n(n + 1)[/B] seen here for the [URL='http://www.mathcelebrity.com/sumofthefirst.php?num=+10&pl=Even+Numbers']first 10 numbers[/URL]

The area of a desert in Africa is 12 times the area of a desert in Asia. If the area of a desert in

The area of a desert in Africa is 12 times the area of a desert in Asia. If the area of a desert in Asia is Y square miles, express the area of a desert in Africa as an algebraic expression in Y.
[B]Africa Area = 12Y[/B]

The coefficient of determination is found by taking the square root of the coefficient of correlatio

The coefficient of determination is found by taking the square root of the coefficient of correlation. True or False
[B]FALSE[/B] - It is found by squaring the coefficient of correlation

The cube of g plus the square of m

The cube of g plus the square of m
The cube of g means we raise g to the 3rd power:
g^3
The square of m means we raise m to the 2nd power:
m^2
The word [I]plus[/I] means we add them both
[B]g^3 + m^2[/B]

The cube of the difference of 5 times the square of y and 7 divided by the square of 2 times y

The cube of the difference of 5 times the square of y and 7 divided by the square of 2 times y
Take this in algebraic expression in parts:
[U]Term 1[/U]
[LIST]
[*]The square of y means we raise y to the 2nd power: y^2
[*]5 times the square of y: 5y^2
[/LIST]
[U]Term 2[/U]
[LIST]
[*]2 times y: 2y
[*]The square of 2 times y: (2y)^2 = 4y^2
[*]7 divide by the square of 2 times y: 7/4y^2
[/LIST]
[U]The difference of these terms is written as Term 1 minus Term 2:[/U]
[LIST]
[*]5y^2/4y^2
[/LIST]
[U]The cube of the difference means we raise the difference to the power of 3:[/U]
[B](5y^2/4y^2)^3[/B]

The difference between the product of 4 and a number and the square of a number

The difference between the product of 4 and a number and the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The product of 4 and a number:
4x
The square of a number means we raise x to the power of 2:
x^2
The difference between the product of 4 and a number and the square of a number:
[B]4x - x^2[/B]

The difference between the square of b and the total of b and 9

The difference between the square of b and the total of b and 9
The square of b means we raise b to the power of 2:
b^2
The total of b and 9 means we add 9 to b:
b + 9
The difference means we subtract:
[B]b^2 - (b + 9)[/B]

The difference between the square of b and the total of d and g

The difference between the square of b and the total of d and g
Square of b means we raise b to the 2nd power:
b^2
Total of d and g:
d + g
The difference between the square of b and the total of d and g
[B]b^2 - (d + g)[/B]

The difference between the squares of two consecutive numbers is 141. Find the numbers

The difference between the squares of two consecutive numbers is 141. Find the numbers
Take two consecutive numbers:
n- 1 and n
Given a difference (d) between the squares of two consecutive numbers, the shortcut for this is:
2n - 1 = d
Proof of this:
n^2- (n - 1)^2 = d
n^2 - (n^2 - 2n + 1) = d
n^2 - n^2 + 2n - 1 = d
2n - 1 = d
Given d = 141, we have
2n - 1 = 141
Add 1 to each side:
2n = 142
Divide each side by 2:
2n/2 = 142/2
n = [B]71[/B]
Therefore, n - 1 = [B]70
Our two consecutive numbers are (70, 71)[/B]
Check your work
70^2 = 4900
71^2 = 5041
Difference = 5041 - 4900
Difference = 141
[MEDIA=youtube]vZJtZyYWIFQ[/MEDIA]

The difference between two positive numbers is 5 and the square of their sum is 169

The difference between two positive numbers is 5 and the square of their sum is 169.
Let the two positive numbers be a and b. We have the following equations:
[LIST=1]
[*]a - b = 5
[*](a + b)^2 = 169
[*]Rearrange (1) by adding b to each side. We have a = b + 5
[/LIST]
Now substitute (3) into (2):
(b + 5 + b)^2 = 169
(2b + 5)^2 = 169
[URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get:
4b^2 + 20b + 25
Set this equal to 169 above:
4b^2 + 20b + 25 = 169
[URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get:
b = (-9, 4)
But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions.
Substitute b = 4 into equation (1) above, and we get:
a - [I]b[/I] = 5
[URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL]
[B]a = 9
[/B]
Therefore, we have [B](a, b) = (9, 4)[/B]

The difference of x and x squared

The difference of x and x squared
We subtract x^2 from x:
[B]x - x^2[/B]

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered t

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
Digit, Probability
1, 0.301
2, 0.176
3, 0.125
4, 0.097
5, 0.079
6, 0.067
7, 0.058
8, 0.051
9, 0.046
[B][U]Fradulent Checks[/U][/B]
Digit, Frequency
1, 36
2, 32
3, 45
4, 20
5, 24
6, 36
7, 15
8, 16
9, 7
Complete parts (a) and (b).
(a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a) What i

The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) What is the probability that a randomly person has an IQ between 85 and 115?
b) Find the 90th percentile of the IQ distribution
c) If a random sample of 100 people is selected, what is the standard deviation of the sample mean?
a) [B]68%[/B] from the [URL='http://www.mathcelebrity.com/probnormdist.php?xone=50&mean=100&stdev=15&n=1&pl=Empirical+Rule']empirical rule calculator[/URL]
b) P(z) = 0.90. so z = 1.28152 using Excel NORMSINV(0.9)

(X - 100)/10 = 1.21852 X = [B]113[/B] rounded up c) Sample standard deviation is the population standard deviation divided by the square root of the sample size 15/sqrt(100) = 15/10 =[B] 1.5[/B]

(X - 100)/10 = 1.21852 X = [B]113[/B] rounded up c) Sample standard deviation is the population standard deviation divided by the square root of the sample size 15/sqrt(100) = 15/10 =[B] 1.5[/B]

The moon's diameter is 2,159 miles. What is the surface area of the moon? Round to the nearest mile.

The moon's diameter is 2,159 miles. What is the surface area of the moon? Round to the nearest mile.
The moon is a sphere. So our Surface Area formula is:
S =4pir^2
If diameter is 2,159, then radius is 2,159/2 = 1079.5. Plug this into the Surface Area of a sphere formula:
S = 4 * pi * 1079.5^2
S = 4 * pi *1165320.25
S = 4661281 pi
S = [B]14,643,846.15 square miles[/B]

The perfect square less than 30

The perfect square less than 30
We know that:
[LIST]
[*]5^2 = 25
[*]6^ = 36
[/LIST]
So our answer is [B]5[/B]

The perimeter of a square with side a

The perimeter of a square with side a
Perimeter of a square is 4s where s is the side length.
With s = a, we have:
P = [B]4a[/B]

The product of a number and its square is less than 8

Let the number be x.
Let the square be x^2.
So we have (x)(x^2) = x^3 < 8
Take the cube root of this, we get x = 2

The quotient of 49 and n squared

n squared is written as n to the power of 2, n^2
We have a fraction, where 49 is the numerator, and n^2 is the denominator
49
-----
n^2

the quotient of m squared and a squared

the quotient of m squared and a squared
[U]m squared means we raise m to the power of 2:[/U]
m^2
[U]a squared means we raise a to the power of 2:[/U]
a^2
[U]The [I]quotient[/I] means we divide m^2 by a^2:[/U]
[B]m^2/a^2[/B]

the quotient of triple m and n squared

the quotient of triple m and n squared
Triple m means we multiply m by 3:
3m
n squared means we raise n to the 2nd power:
n^2
The quotient is formed as follows:
[B]3m/n^2[/B]

The square of a number added to its reciprocal

The square of a number added to its reciprocal
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
the square of x mean we raise x to the power of 2. It's written as:
x^2
The reciprocal of x is 1/x
We add these together to get our final algebraic expression:
[B]x^2 + 1/x
[MEDIA=youtube]ZHut58-AoDU[/MEDIA][/B]

The square of a number increased by 7 is 23

The square of a number increased by 7 is 23
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
The square of a number means we raise x to the power of 2:
x^2
[I]Increased by[/I] means we add 7 to x^2
x^2 + 7
The word [I]is[/I] means an equation. So we set x^2 + 7 equal to 23:
[B]x^2 + 7 = 23[/B]

The square of a number is always nonnegative.

The square of a number is always nonnegative.
This is true, and here is why:
Suppose you have a positive number n.
n^2 = n * n
A positive times a positive is a positive
Suppose you have a negative number -n
(-n)^2 = -n * -n = n^2
A negative times a negative is a positive.

The square of a number is positive

The square of a number is positive
N ca be positive or negative, so test both scenarios:
Take a positive number n.
n^2 = n^2 * n^2 or Positive * Positive which is positive
Take a negative number n
(-n)^2 = -n * -n or Negative * Negative which is positive
(-n)^2 = n^2

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 square of a positive integer minus twice its consecutive integer is equal to 22. find the intege

The square of a positive integer minus twice its consecutive integer is equal to 22. Find the integers.
Let x = the original positive integer. We have:
[LIST]
[*]Consecutive integer is x + 1
[*]x^2 - 2(x + 1) = 22
[/LIST]
Multiply through:
x^2 - 2x - 2 = 22
Subtract 22 from each side:
x^2 - 2x - 24 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2-2x-24%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get:
x = 6 and x = -4
Since the problem states [U]positive integers[/U], we use:
x = 6 and x + 1 = 7
[B](6, 7)[/B]

The square of the difference of a number and 4

The square of the difference of a number and 4
A number means an arbitrary variable, let's call it x
The difference of a number and 4:
x - 4
The square of this difference:
[B](x - 4)^2[/B]

The square of the difference of n and 2, increased by twice n

The square of the difference of n and 2, increased by twice n
The difference of n and 2:
n - 2
The square of the difference of n and 2 means we raise (n - 2) to the 2nd power:
(n - 2)^2
Twice n means we multiply n by 2:
2n
The square of the difference of n and 2, increased by twice n
[B](n - 2)^2 + 2n[/B]

The square of the radius r

The square of the radius r
The square means you raise r to the power of 2:
[B]r^2[/B]

the square of the sum of 2a and 3b

the square of the sum of 2a and 3b
the sum of 2a and 3b
2a + 3b
The square of this sum means we raise 2a + 3b to the 2nd power:
[B](2a + 3b)^2[/B]

the square of the sum of p and 5

the square of the sum of p and 5
The sum of p and 5
p + 5
Square this sum:
[B](p + 5)^2[/B]

The square of the sum of twice a number x and y

The square of the sum of twice a number x and y
Take this in algebraic expression in 3 parts:
[LIST=1]
[*]Twice a number x means we multiply x by 2: 2x
[*]The sum of twice a number x and y means we add y to 2x above: 2x + y
[*]The square of the sum means we raise the sum (2x + y) to the second power below:
[/LIST]
[B](2x + y)^2[/B]

the square of the sum of two numbers

the square of the sum of two numbers
Let the first number be x. Let the second number be y.
The sum is:
x + y
Now we square that sum by raising the sum to a power of 2:
[B](x + y)^2[/B]

the square of the sum of x and y is less than 20

the square of the sum of x and y is less than 20
The sum of x and y means we add y to x:
x + y
the square of the sum of x and y means we raise the term x + y to the 2nd power:
(x + y)^2
The phrase [I]is less than[/I] means an inequality, so we write this as follows:
[B](x + y)^2 < 20[/B]

the square root of twice a number is 4 less than the number

Write this out, let the number be x.
sqrt(2x) = x - 4 since 4 less means subtract
Square each side:
sqrt(2x)^2 = (x - 4)^2
2x = x^2 - 8x + 16
Subtract 2x from both sides
x^2 - 10x + 16 = 0
Using the [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2+-+10x+%2B+16+%3D+0&pl=Solve+Quadratic+Equation&hintnum=0']quadratic calculator[/URL], we get two potential solutions
x = (2, 8)
Well, 2 does not work, since sqrt(2*2) = 2 which is not 4 less than 2
However, 8 does work:
sqrt(2*8) = sqrt(16) = 4, which is 4 less than the number 8.

the sum of 16 squared and a number x

the sum of 16 squared and a number x
16 squared:
16^2
The sum of this and a number x
[B]x + 16^2[/B]

The sum of 3 times the square of a number and negative 7

The sum of 3 times the square of a number and negative 7
[U]The phrase [I]a number[/I] means an arbitrary variable, let's call it x:[/U]
x
[U]The square of a number means we raise x to the power of 2:[/U]
x^2
[U]3 times the square of a number:[/U]
3x^2
[U]The sum of 3 times the square of a number and negative 7[/U]
[B]3x^2 - 7[/B]

The sum of a number and its square is 72. find the numbers?

The sum of a number and its square is 72. find the numbers?
Let the number be n. We have:
n^2 + n = 72
Subtract 72 from each side:
n^2 + n - 72 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we have:
[B]n = 8 or n = -9
[/B]
Since the numbers do not state positive or negative, these are the two solutions.

The sum Of a number squared and 14

The sum Of a number squared and 14.
A number means an arbitrary variable, let's call it x.
Squared means we raise x to the 2nd power: x^2
The sum means we add x^2 to 14 to get our algebraic expression below:
[B]x^2 + 14[/B]

The sum of the square of a number and 7 times a number

The sum of the square of a number and 7 times a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Square the number:
x^2
7 times the number means we multiply x by 7:
7x
The sum means we add x^2 and 7x
[B]x^2 + 7x[/B]

the sum of the squares of a and b

the sum of the squares of a and b
Square of a means we raise a to the 2nd power:
a^2
Square of b means we raise b to the 2nd power:
b^2
The sum of squares means we add these terms together to get our algebraic expression:
[B]a^2 + b^2[/B]

The sum of the squares of c and d is 25

The sum of the squares of c and d is 25
The square of c means we we raise c to the power of 2:
c^2
The square of d means we we raise d to the power of 2:
d^2
The sum of the squares of c and d means we add d^2 to c^2:
c^2 + d^2
The word [I]is[/I] means equal to, so we set c^2 + d^2 equal to 25:
[B]c^2 + d^2 = 25[/B]

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.
Let the 2 consecutive integers be x and x + 1. We have:
x^2 + (x + 1)^2 = 61
Simplify:
x^2 + x^2 + 2x + 1 = 61
2x^2 + 2x + 1 = 61
Subtract 61 from each side:
2x^2 + 2x - 60 = 0
Divide each side by 2
x^2 + x - 30
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL], we get:
x = 5 and x = -6
The question asks for [I]positive integers[/I], so we use [B]x = 5. [/B]This means the other number is [B]6[/B].

the sum of x squared plus y squared

the sum of x squared plus y squared
x squared means we raise x to the power of 2:
x^2
y squared means we raise y to the power of 2:
y^2
The sum means we add both terms together:
[B]x^2 + y^2[/B]

the total of 3 times the cube of u and the square of u

the total of 3 times the cube of u and the square of u
[U]The cube of u means we raise u to the power of 3:[/U]
u^3
[U]The square of u means we raise u to the power of 2:[/U]
u^2
The total of both of these is found by adding them together:
[B]u^3 + u^2[/B]

total of a number and the square of a number

total of a number and the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The square of a number means we raise x to the power of 2. x^2
The total means we add x squared to x:
[B]x + x^2[/B]

Trinomials

Free Trinomials Calculator - Checks to see if equations in the trinomial form ax^{2} + bx + c are a perfect square as well as completing the square from equations in the form ax^{2} + bx + ?.
Also shows you a perfect square trinomial

tripled square of the difference of a and b

The difference of a and b is written as:
a - b
Square the difference means raise the difference to the power of 2
(a - b)^2
Triple this expression means multiply by 3:
[B]3(a - b)^2[/B]

True or False: The standard deviation of the chi-square distribution is twice the mean.

True or False: The standard deviation of the chi-square distribution is twice the mean.
[B]False[/B], the variance is twice the mean. Mean is k, Variance is 2k

twice a number subtracted from the square root of the same number

twice a number subtracted from the square root of the same number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Twice a number means we multiply x by 2:
2x
Square root of the same number:
sqrt(x)
twice a number subtracted from the square root of the same number
[B]sqrt(x) - 2x[/B]

twice the square of the product of x and y

twice the square of the product of x and y
Take this algebraic expression in pieces:
[LIST]
[*]The product of x and y means we multiply x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2 = x^2y^2
[*]Twice the square means we multiply the square by 2: [B]2x^2y^2[/B]
[/LIST]

twice the square of the product of x and y

twice the square of the product of x and y
[LIST]
[*]The product of x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2
[*]Twice the square means we multiply by 2
[/LIST]
[B]2(xy)^2
or
2x^2y^2[/B]

twice the square root of a number increased by 5 is 23

twice the square root of a number increased by 5 is 23
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The square root of a number means we raise x to the 1/2 power:
sqrt(x)
the square root of a number increased by 5 means we add 5 to sqrt(x):
sqrt(x) + 5
twice the square root of a number increased by 5 means we multiply sqrt(x) + 5 by 2:
2(sqrt(x) + 5)
The phrase [I]is 23[/I] means we set 2(sqrt(x) + 5) equal to 23:
[B]2(sqrt(x) + 5) = 23[/B]

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.
If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.
The sum of their squares is:
x^2+(20 - x)^2
Expand this and we get:
x^2 + 400 - 40x + x^2
Combine like terms:
2x^2 - 40x + 400
Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x−10)^2 + 200
The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.
The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10
Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.
If x = 10, then the other number is 20 - 10 = 10.

u varies jointly as q and the square of m

u varies jointly as q and the square of m
Varies jointly means we multiply. There exists a constant k such that:
[B]u = kqm^2[/B]

Use k as the constant of variation. L varies jointly as u and the square root of v.

Use k as the constant of variation. L varies jointly as u and the square root of v.
Since u and v vary jointly, we multiply by the constant of variation k:
[B]l = ku * sqrt(v)[/B]

Variation Equations

Free Variation Equations Calculator - This calculator solves the following direct variation equations and inverse variation equations below:

* y varies directly as x

* y varies inversely as x

* y varies directly as the square of x

* y varies directly as the cube of x

* y varies directly as the square root of x

* y varies inversely as the square of x

* y varies inversely as the cube of x

* y varies inversely as the square root of x

* y varies directly as x

* y varies inversely as x

* y varies directly as the square of x

* y varies directly as the cube of x

* y varies directly as the square root of x

* y varies inversely as the square of x

* y varies inversely as the cube of x

* y varies inversely as the square root of x

vw^2+y=x for w

vw^2+y=x for w
This is an algebraic expression.
Subtract y from each side:
vw^2 + y - y = x - y
The y's cancel on the left side, so we're left with:
vw^2 = x - y
Divide each side by v
w^2 = (x - y)/v
Take the square root of each side:
w = [B]Sqrt((x - y)/v)[/B]

What is the ratio of the area of a circle to the area of a square drawn around that circle? Express

What is the ratio of the area of a circle to the area of a square drawn around that circle? Express your answer in terms of pi.
Area of a circle = pir^2
area of a square = (2r)^2 = 4r^2
Ratio = pir^2/4r^2
Ratio = [B]pi/4[/B]

What is the value of 998^2 – 2^2?

A) 988,036
B) 990,000
C) 995,988
D) 996,000
E) 1,000,000
This is a difference of squares.
The formula for 2 numbers a and b is:
a^2 - b^2 = (a + b)(a - b)
In our problem, we have a = 998 and b = 2:
998^2 – 2^2 = (998 + 2)(998 - 2)
998^2 – 2^2 = 1000(996)
Multiplying by 1000 means we move the decimal place of the other number 3 places to the right:
998^2 – 2^2 = [B]996,000 or Answer D
[MEDIA=youtube]IeKLs8Ds-No[/MEDIA][/B]

what two values can d have if d squared is 9

what two values can d have if d squared is 9
d^2 = 9
Using [URL='https://www.mathcelebrity.com/radex.php?num=sqrt(9%2F1)&pl=Simplify+Radical+Expression']our calculator[/URL], we get:
d = [B](-3, 3}[/B]

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negati

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution.
Let the number be n.
Square of a number:
n^2
28 is subtracted from the square of a number:
n^2 - 28
3 times the number:
3n
[I]The result is[/I] mean an equation, so we set n^2 - 28 = 3n
n^2 - 28 = 3n
Subtract 3n from each side:
n^2 - 3n - 28 = 3n - 3n
The right side cancels to 0, so we have:
n^2 - 3n - 28 = 0
This is a quadratic equation in standard form, so we [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-28%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']use our quadratic calculator[/URL] to solve:
We get two solutions for n:
n = (-4, 7)
The question asks for the negative solution, so our answer is:
[B]n = -4[/B]

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positiv

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution.
Let the number be n. We have:
n^2 - 4 = 3n
Subtract 3n from each side:
n^2 - 3n - 4 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Typing this quadratic equation into the search engine[/URL], we get:
n = (-1, 4)
The problem asks for the positive solution, so we get [B]n = 4[/B].

When 54 is subtracted from the square of a number, the result is 3 times the number.

When 54 is subtracted from the square of a number, the result is 3 times the number.
This is an algebraic expression. Let's take it in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it "x".
x
Square the number, means raise it to the 2nd power:
x^2
Subtract 54:
x^2 - 54
The phrase [I]the result[/I] means an equation, so we set x^2 - 54 equal to 3
[B]x^2 - 54 = 3[/B]

When the side of a square is doubled in length, its area increases by 432 square inches. What is the

When the side of a square is doubled in length, its area increases by 432 square inches. What is the size of the original square?
Original square side length is s
Area = s^2
Double the side lengths to 2s
New area = (2s)^2 = 4s^2
Setup the difference relation:
4s^2 - s^2 = 432
3s^2 = 432
Divide each side by 3:
3s^2/3 = 432/3
s^2 = 144
s = [B]12[/B]

x cubed plus x squared decreased by 7

x cubed plus x squared decreased by 7
[U]x cubed means we raise x to the power of 3:[/U]
x^3
[U]x squared means we raise x to the power of 2:[/U]
x^2
[U]x cubed plus x squared[/U]
x^3 + x^2
[U]Decreased by 7:[/U]
[B]x^3 + x^2 - 7[/B]

x squared plus a minus b

x squared plus a minus b
x squared means we raise x to the power of 2:
x^2
Plus a:
x^2 + a
Minus b:
[B]x^2 + a - b[/B]

x squared times the difference of x and y

x squared times the difference of x and y
x squared means we raise x to the power of 2:
x^2
The difference of x and y
x - y
x squared times the difference of x and y
[B]x^2(x - y)[/B]

You and your friend are playing a number-guessing game. You ask your friend to think of a positive n

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]

You are purchasing carpeting for an office building. The space to be carpeted is 30 feet by 50 feet.

You are purchasing carpeting for an office building. The space to be carpeted is 30 feet by 50 feet. Company A charges $2.99 per square foot plus a $200 installation charge. Company B charges $19.99 per square yard plus a $500 installation charge. What is the best deal?
Did you notice the word snuck in on this problem? Company B is given in square [I][B]yards[/B][/I], not feet. Let's convert their price to square feet to match company A.
[U]Company B conversion:[/U]
Since we have 1 square yard = 3 feet * 3 feet = 9 square feet, we need to solve the following proportion:
$19.99/square yard * 1 square yard/9 feet = $19.99 square yard / 9 feet = $2.22 / square foot.
Now, let's set up the cost equations C(s) for each Company in square feet (s)
[LIST]
[*]Company A: C(s) = 200 + 2.99s
[*]Company B: C(s) = 500 + 2.22s
[/LIST]
The problem asks for s = 30 feet * 50 feet = 1500 square feet. So we want to calculate C(1500)
[U]Company A:[/U]
C(1500) = 200 + 2.99(1500)
C(1500) = 200 + 4485
C(1500) = 4685
[U]Company B:[/U]
C(1500) = 500 + 2.22(1500)
C(1500) = 500 + 3330
C(1500) = 3830
Since [B]Company B[/B] has the lower cost per square foot, they are the better buy.

z is directly proportional to the square of x and y

z is directly proportional to the square of x and y
Directly proportional means there exists a constant k such that:
z = [B]kx^2y
[MEDIA=youtube]J3ByZkcX38E[/MEDIA][/B]

z is jointly proportional to the square of x and the cube of y

z is jointly proportional to the square of x and the cube of y
The square of x means we raise x to the power of 2:
x^2
The cube of y means we raise y to the power of 3:
y^3
The phrase [I]jointly proportional[/I] means we have a constant k such that:
[B]z = kx^2y^3[/B]

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10
Varies inversely means there exists a constant k such that:
z = k/t^2
If z = 4 when t = 2, we have:
4 = k/2^2
4 = k/4
Cross multiply and we get:
k = 4 * 4
k = 16
Now the problem asks to find z when t is 10:
z = k/t^2
z = 16/10^2
z = 16/100
z = [B]0.16[/B]