distance - interval between two points in time

Formula: d = rt

3-dimensional points

Calculates distance between two 3-dimensional points

(x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) as well as the parametric equations and symmetric equations

(x

35 m/s for 40 s. how far does it travel?

35 m/s for 40 s. how far does it travel?
This is a distance problem. The formula to relate, distance, rate, and time is:
d = rt
We are given r = 35 m/s and t = 40s. We want d
d = 35 m/s * 40s
d = [B]1,400 meters[/B]

6 mph, 2 hours what is the distance

6 mph, 2 hours what is the distance
Distance = Rate * Time
Distance = 6 mph * 2 hours
Distance = [B]12 miles
[/B]
You can also use our [URL='http://www.mathcelebrity.com/drt.php?d=+&r=+6&t=+2&pl=Calculate+the+missing+Item+from+D%3DRT']distance-rate-time calculator[/URL]

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an ho

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?
[U]Set up the relationship of still water speed and downstream speed[/U]
Speed down stream = Speed in still water + speed of the current
Speed down stream = x+2
Therefore:
Speed upstream =x - 2
Since distance = rate * time, we rearrange to get time = Distance/rate:
15/(x+ 2) + 15 /(x- 2) = 3
Multiply each side by 1/3 and we get:
5/(x + 2) + 5/(x - 2) = 1
Using a common denominator of (x + 2)(x - 2), we get:
5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2)
(5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2)
10x = (x+2)(x-2)
We multiply through on the right side to get:
10x = x^2 - 4
Subtract 10x from each side:
x^2 - 10x - 4 = 0
This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get:
Speed of the boat in still water =X=5 +- sq. Root of 29 kmph
We only want the positive solution:
x = 5 + sqrt(29)
x = 10.38
[U]Calculate time for upstream journey:[/U]
Time for upstream journey = 15/(10.38 - 2)
Time for upstream journey = 15/(8.38)
Time for upstream journey = [B]1.79[/B]
[U]Calculate time for downstream journey:[/U]
Time for downstream journey = 15/(10.38 + 2)
Time for downstream journey = 15/(12.38)
Time for downstream journey = [B]1.21[/B]

A ball is dropped from a height of 12 feet and returns to a height that is one-half of the height fr

A ball is dropped from a height of 12 feet and returns to a height that is one-half of the height from which it fell. The ball continues to bounce half the height of the previous bounce each time. How far will the ball have traveled when it hits the ground for the fifth time?
Take the top of the bounces one at a time:
[LIST=1]
[*]Ball is dropped 12 feet and it bounces up to 6 feet
[*]Ball drops 6 feet back down and bounces up to 3 feet up
[*]Ball drops 3 feet back down and bounces up to 1.5 feet up
[*]Ball drops 1.5 feet down and bounces up to 0.75 feet up
[*]Bounce 5 is 0.75 feet down
[/LIST]
[U]Total distance travelled:[/U]
12 + 6 + 6 + 3 + 3 + 1.5 + 1.5 + 0.75
[B]33.75 feet[/B]

A ball was dropped from a height of 6 feet and began bouncing. The height of each bounce was three-f

A ball was dropped from a height of 6 feet and began bouncing. The height of each bounce was three-fourths the height of the previous bounce. Find the total vertical distance travelled by the all in ten bounces.
The height of each number bounce (n) is shown as:
h(n) = 6(0.75)^n
We want to find h(10)
h(n) = 6(0.75)^n
Time Height
0 6
1 4.5
2 3.375
3 2.53125
4 1.8984375
5 1.423828125
6 1.067871094
7 0.8009033203
8 0.6006774902
9 0.4505081177
10 0.3378810883
Adding up each bounce from 1-10, we get:
16.98635674
Since vertical distance means both [B]up and down[/B], we multiply this number by 2 to get:
16.98635674 * 2 = 33.97271347
Then we add in the initial bounce of 6 to get:
33.97271347 + 6 = [B]39.97271347 feet[/B]

a boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equa

a boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equation to describe this relationship
We know the distance (d) equation in terms of rate (r) and time (t) as:
d = rt
We're given d = 336km and t = 12 hours, so we have:
[B]336 km = 12t [/B] <-- this is our equation
Divide each side by 12 to solve for t:
12t/12 = 336/12
t = [B]28 km / hour[/B]

A boat traveled at a constant speed for 32 hours, covering a total distance of 597 kilometers. To th

A boat traveled at a constant speed for 32 hours, covering a total distance of 597 kilometers. To the nearest hundredth of a kilometer per hour, how fast was it going?
Distance = Rate * Time
We're given t = 32, and d = 597. Using our [URL='https://www.mathcelebrity.com/drt.php?d=+597&r=+&t=32&pl=Calculate+the+missing+Item+from+D%3DRT']distance, rate, and time calculator[/URL], we get:
r = [B]18.656 km/hr[/B]

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

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

A car is traveling 60 km per hour. How many hours will it take for the car to reach a point that is

A car is traveling 60 km per hour. How many hours will it take for the car to reach a point that is 180 km away?
Rate * Time = Distance so we have t for time as:
60t = 180
To solve this equation for t, we [URL='https://www.mathcelebrity.com/1unk.php?num=60t%3D180&pl=Solve']type it in the search engine[/URL] and we get:
t = [B]3[/B]

A car travels 16 m/s and travels 824 m. How long was the car moving?

A car travels 16 m/s and travels 824 m. How long was the car moving?
Distance = Rate * Time, so we have:
824m = 16m/s * t
Using our [URL='https://www.mathcelebrity.com/drt.php?d=+824&r=16&t=&pl=Calculate+the+missing+Item+from+D%3DRT']distance calculator[/URL], we get:
[B]51.5 seconds[/B]

A car travels 71 feet each second.How many feet does it travel in 12 seconds?

A car travels 71 feet each second.How many feet does it travel in 12 seconds?
Distance = Rate * Time
We're given a rate of 71 feet per second and a time of 12 seconds. So we plug this in:
Distance = 71 feet/second * 12 seconds
[URL='https://www.mathcelebrity.com/drt.php?d=+&r=71&t=+12&pl=Calculate+the+missing+Item+from+D%3DRT']Distance[/URL] = [B]852 feet[/B]

A car travels at 40 kilometers per hour. Write an expression for the distance traveled after h hours

A car travels at 40 kilometers per hour. Write an expression for the distance traveled after h hours.
Distance = rate * time, so we have:
Distance = 40km/h * h
Distance = [B]40h[/B]

A certain race is a distance of 26 furlongs. How far is the race in (a) miles? (b) yards?

A certain race is a distance of 26 furlongs. How far is the race in (a) miles? (b) yards?
[URL='https://www.mathcelebrity.com/linearcon.php?quant=26&pl=Calculate&type=furlong']We type in [I]26 furlongs[/I] into our search engine[/URL] and we get:
[LIST]
[*][B]3.25 miles[/B]
[*][B]5,720 yards[/B]
[/LIST]

A cheetah can maintain it's maximum speed of 28 m/s for 30 seconds. how far does it go in this amoun

A cheetah can maintain it's maximum speed of 28 m/s for 30 seconds. how far does it go in this amount of time?
Distance = rate * time
Distance = 28 m/s * 30 s
Distance = [B]840m[/B]

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

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

A circle has a center at (6, 2) and passes through (9, 6)

A circle has a center at (6, 2) and passes through (9, 6)
The radius (r) is found by [URL='https://www.mathcelebrity.com/slope.php?xone=6&yone=2&slope=+2%2F5&xtwo=9&ytwo=6&pl=You+entered+2+points']using the distance formula[/URL] to get:
r = 5
And the equation of the circle is found by using the center (h, k) and radius r as:
(x - h)^2 + (y - k)^2 = r^2
(x - 6)^2 + (y - 2)^2 = 5^2
[B](x - 6)^2 + (y - 2)^2 = 25[/B]

A driver drove at a speed of 42 mph for z hours. How far did the driver go?

A driver drove at a speed of 42 mph for z hours. How far did the driver go?
Distance = Rate * Time, so we have:
Distance = [B]42z[/B]

A driver drove at a speed of 56 mph for z hours. How far did the driver go?

A driver drove at a speed of 56 mph for z hours. How far did the driver go?
Distance = Rate * time
So we have:
Distance = 56 mph * z
Distance = [B]56z[/B]

A driver drove at a speed of 58 mph for t hours. How far did the driver go?

A driver drove at a speed of 58 mph for t hours. How far did the driver go?
Since distance = rate * time, we have distance D of:
[B]D = 58t[/B]

A family is taking a cross-country trip of 3000 miles by car. They are bringing two spare tires with

A family is taking a cross-country trip of 3000 miles by car. They are bringing two spare tires with them and want all six tires to go an equal distance. How many miles will each tire go?
3000 * 4 tires = 12,000 miles traveled
12,000 / 6 tires = [B]2,000 miles[/B]

A girl is pedaling her bicycle at a velocity of 0.10 km/hr. How far will she travel in two hours?

A girl is pedaling her bicycle at a velocity of 0.10 km/hr. How far will she travel in two hours?
The distance formula is:
d = rt
We're given a rate (r) of 0.10km/hr
We're given time (t) of 2 hours
Plug these values into the distance formula and we get:
d= 0.1 * 2
d = [B]0.2km
[MEDIA=youtube]w80E_YM-tDA[/MEDIA][/B]

A helicopter rose vertically 300 m and then flew west 400 m how far was the helicopter from it’s sta

A helicopter rose vertically 300 m and then flew west 400 m how far was the helicopter from it’s starting point?
The distance forms a right triangle. We want the distance of the hypotenuse.
Using our [URL='http://www.mathcelebrity.com/pythag.php?side1input=300&side2input=400&hypinput=&pl=Solve+Missing+Side']right triangle calculator[/URL], we get a distance of [B]500[/B].
We also could use a shortcut on this problem. If you divide 300 and 400 by 100, you get 3 and 4. Since we want the hypotenuse, you get the famous 3-4-5 triangle ratio. So the answer is 5 * 100 = 500.

A jet left Nairobi and flew east at an average speed of 231 mph. A passenger plane left four hours l

A jet left Nairobi and flew east at an average speed of 231 mph. A passenger plane left four hours later and flew in the same direction but with an average speed of 385 mph. How long did the jet fly before the passenger plane caught up?
Jet distance = 231t
Passenger plane distance = 385(t - 4)
385(t - 4) = 231t
385t - 1540 = 231t
Subtract 231t from each side
154t = 1540
[URL='https://www.mathcelebrity.com/1unk.php?num=154t%3D1540&pl=Solve']Type 154t = 1540[/URL] into the search engine, we get [B]t = 10.
[/B]
Check our work:
Jet distance = 231(10) = 2,310
Passenger plane distance = 385(10 - 4) = 385 * 6 = 2,310

A jet plane traveling at 550 mph over takes a propeller plane traveling at 150 mph that had a 3 hour

A jet plane traveling at 550 mph over takes a propeller plane traveling at 150 mph that had a 3 hours head start. How far from the starting point are the planes?
Use the formula D = rt where
[LIST]
[*]D = distance
[*]r = rate
[*]t = time
[/LIST]
The plan traveling 150 mph for 3 hours:
Time 1 = 150
Time 2 = 300
Time 3 = 450
Now at Time 3, the other plane starts
Time 4 = 600
Time 5 = 750
Time 6 =
450 + 150t = 550t
Subtract 150t
400t = 450
Divide each side by 400
t = 1.125
Plug this into either distance equation, and we get:
550(1.125) = [B]618.75 miles[/B]

A jet travels 832 km in 5 hours. At this rate, how far could the jet fly in 12 hours? What is the ra

A jet travels 832 km in 5 hours. At this rate, how far could the jet fly in 12 hours? What is the rate of speed of the jet?
Distance = rate * time. We're given D = 832 and t = 5. Using our [URL='https://www.mathcelebrity.com/drt.php?d=+832&r=+&t=+5&pl=Calculate+the+missing+Item+from+D%3DRT']drt calculator[/URL], we solve or rate to get:
[B]r = 166.4[/B]
The problems asks for a distance D when t = 12 hours and r = 166.4 from above. Using our [URL='https://www.mathcelebrity.com/drt.php?d=&r=+166.4&t=+12&pl=Calculate+the+missing+Item+from+D%3DRT']drt calculator solving for d[/URL], we get:
d = [B]1,996.8 km[/B]

A jet travels at 485 miles per hour. Which equation represents the distance, d, that the jet will tr

A jet travels at 485 miles per hour. Which equation represents the distance, d, that the jet will travel in t hours.
The distance formula is:
d = rt
We're given r = 485, so we have:
[B]d = 485t[/B]

A meter is defined as the distance light travels in 1/299,792,458 of a second. How many meters does

A meter is defined as the distance light travels in 1/299,792,458 of a second. How many meters does light travel in 1/8 of a second?
1/8 second / 1/299,792,458
299,792,458/8 = [B]37,474,057.25 meters[/B]

A motorboat travels 408 kilometers in 8 hours going upstream and 546 kilometers in 6 hours going dow

A motorboat travels 408 kilometers in 8 hours going upstream and 546 kilometers in 6 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?
[U]Assumptions:[/U]
[LIST]
[*]B = the speed of the boat in still water.
[*]S = the speed of the stream
[/LIST]
Relative to the bank, the speeds are:
[LIST]
[*]Upstream is B - S.
[*]Downstream is B + S.
[/LIST]
[U]Use the Distance equation: Rate * Time = Distance[/U]
[LIST]
[*]Upstream: (B-S)6 = 258
[*]Downstream: (B+S)6 = 330
[/LIST]
Simplify first by dividing each equation by 6:
[LIST]
[*]B - S = 43
[*]B + S = 55
[/LIST]
Solve this system of equations by elimination. Add the two equations together:
(B + B) + (S - S) = 43 + 55
Cancelling the S's, we get:
2B = 98
Divide each side by 2:
[B]B = 49 mi/hr[/B]
Substitute this into either equation and solve for S.
B + S = 55
49 + S = 55
To solve this, we [URL='https://www.mathcelebrity.com/1unk.php?num=49%2Bs%3D55&pl=Solve']type it in our search engine[/URL] and we get:
S = [B]6 mi/hr[/B]

A parabola has a Vertex at (4,-2) and a Focus at (6,-2). Find the equation of the parabola

A parabola has a Vertex at (4,-2) and a Focus at (6,-2). Find the equation of the parabola and the lotus rectum.
Equation of a parabola given the vertex and focus is:
([I]x[/I] – [I]h[/I])^2 = 4[I]p[/I]([I]y[/I] – [I]k[/I])
The vertex (h, k) is 4, -2
The distance is p, and since the y coordinates of -2 are equal, the distance is 6 - 4 = 2.
So p = 2
Our parabola equation becomes:
(x - 4)^2 = 4(2)(y - -2)
[B](x - 4)^2 = 8(y + 2)[/B]
Latus rectum of a parabola is 4p, where p is the distance between the vertex and the focus
LR = 4p
LR = 4(2)
[B]LR = 8[/B]

A passenger train left station A at 6:00 P.M. Moving with the average speed 45 mph, it arrived at st

A passenger train left station A at 6:00 P.M. Moving with the average speed 45 mph, it arrived at station B at 10:00 p.m. A transit train left from station A 1 hour later than the passenger train, but it arrived at the station B at the same time with the passenger train. What was the average speed of the transit train?
[U]Passenger Train[/U]
[LIST]
[*]45 miles per hour and it got there in 4 hours.
[/LIST]
Using our formula D = rt where:
[LIST]
[*]D = Distance
[*]r = rate
[*]t = time
[/LIST]
[LIST]
[*]D = rt
[*]D = 45(4)
[*]D = 180 miles from Station A to Station B
[/LIST]
Transit Train
[LIST]
[*]It has to go the same distance, 180 miles, so D = 180
[*]It made it there in 3 hours. This is r
[*]We want to solve for t
[/LIST]
D = rt
180 = 3r
Divide each side by 3
[B]r = 60 miles per hour[/B]

A police officer is trying to catch a fleeing criminal. The criminal is 20 feet away from the cop, r

A police officer is trying to catch a fleeing criminal. The criminal is 20 feet away from the cop, running at a rate of 5 feet per second. The cop is running at a rate of 6.5 feet per second. How many seconds will it take for the police officer to catch the criminal?
Distance = Rate * Time
[U]Criminal:[/U]
5t + 20
[U]Cop[/U]:
6.5t
We want to know when their distances are the same (cop catches criminal). So we set the equations equal to each other:
5t + 20 = 6.5t
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=5t%2B20%3D6.5t&pl=Solve']we type it in our search engine[/URL] and we get:
t = 13.333 seconds

A private jet flies the same distance in 4 hours that a commercial jet flies in 2 hours. If the spee

A private jet flies the same distance in 4 hours that a commercial jet flies in 2 hours. If the speed of the commercial jet was 154 mph less than 3 times the speed of the private jet, find the speed of each jet.
Let p = private jet speed and c = commercial jet speed. We have two equations:
(1) c = 3p - 154
(2) 4p =2c
Plug (1) into (2):
4p = 2(3p - 154)
4p = 6p - 308
Subtract 4p from each side:
2p - 308 = 0
Add 308 to each side:
2p = 308
Divide each side by 2:
[B]p = 154[/B]
Substitute this into (1)
c = 3(154) - 154
c = 462 - 154
[B]c = 308[/B]

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

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

A road construction team built a 114 mile road over a period of 19 months what was their average bui

A road construction team built a 114 mile road over a period of 19 months what was their average building distance per a month
Average building distance = miles built / months of building
Average building distance = 114/19
Average building distance = [B]6 miles per month[/B]

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

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

A straight road to the top of a hill is 2500 feet long and makes an angle of 12 degrees with the hor

A straight road to the top of a hill is 2500 feet long and makes an angle of 12 degrees with the horizontal. Find the height of the hill.
Height = Distance * Sin(Horizon Angle)
Height = 2500 * [URL='http://www.mathcelebrity.com/anglebasic.php?entry=12&coff=&pl=sin']Sin(12)[/URL]
Height = 2500 * 0.207911691
Height = [B]519.78 feet[/B]

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

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

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
Set up the travel cost equation where m is the number of miles:
C(m) = 0.8m + 1.50
If Samantha wants to spend less than 12 per ride, we have an inequality where C(m) < 12:
[B]0.8m + 1.50 < 12[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
[LIST]
[*]Each ride will cost 1.50 + 0.8x where x is the number of miles per trip.
[*]This expression must be less than 12.
[/LIST]
[U]Setup the inequality:[/U]
1.5 + 0.8x < 12
[U]Subtracting 1.5 from each side of the inequality[/U]
0.8x < 10.5
[U]Simplifying even more by dividing each side of the inequality by 0.8, we have:[/U]
[B]x < 13.125[/B]

A tortoise is walking in the desert. It walks at a speed of 5 meters per minute for 12.5 meters. For

A tortoise is walking in the desert. It walks at a speed of 5 meters per minute for 12.5 meters. For how many minutes does it walk?
Distance formula (d) for a rate (r) and time (t) is:
d = rt
We're given d = 12.5 and r = 5
12.5 = 5t
5t = 12.5
Solve for t. Divide each side of the equation by 5:
5t/5 = 12.5/5
Cancel the 5's on left side and we get:
t = [B]2.5[/B]

A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was

A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was Marcos car towed if he received a bill for $71
Set up a cost equation C(m) where m is the number of miles:
C(m) = Cost per mile * m + Service Fee
Plugging in the service fee of 50 and cost per mile of 1.75, we get:
C(m) = 1.75m + 50
The question asks for what m is C(m) = 71. So we set C(m) = 71 and solve for m:
1.75m + 50 = 71
Solve for [I]m[/I] in the equation 1.75m + 50 = 71
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 50 and 71. To do that, we subtract 50 from both sides
1.75m + 50 - 50 = 71 - 50
[SIZE=5][B]Step 2: Cancel 50 on the left side:[/B][/SIZE]
1.75m = 21
[SIZE=5][B]Step 3: Divide each side of the equation by 1.75[/B][/SIZE]
1.75m/1.75 = 21/1.75
m = [B]12[/B]

A tractor tire has a radius of 24 inches. If the tire rotates one time around, about how many inches

A tractor tire has a radius of 24 inches. If the tire rotates one time around, about how many inches of ground will it cover? Use 3.14 for pi.
A tractor tire is a circle. We want the circumference, which is the distance around the tire.
C = 2pir
C = 2(3.1415)24
[B]C ~ 150.8[/B]

A train leaves San Diego at 1:00 PM. A second train leaves the same city in the same direction at 3

A train leaves San Diego at 1:00 PM. A second train leaves the same city in the same direction at 3:00 PM. The second train travels 30mph faster than the first. If the second train overtakes the first at 6:00 PM, what is the speed of each of the two trains?
Distance = Rate x Time
Train 1:
d = rt
t = 1:oo PM to 6:00 PM = 5 hours
So we have d = 5r
Train 2:
d = (r + 30)t
t = 3:oo PM to 6:00 PM = 3 hours
So we have d = 3(r + 30)
Set both distances equal to each other since overtake means Train 2 caught up with Train 1, meaning they both traveled the same distance:
5r = 3(r + 30)
Multiply through:
3r + 90 = 5r
[URL='https://www.mathcelebrity.com/1unk.php?num=3r%2B90%3D5r&pl=Solve']Run this equation through our search engine[/URL], and we get [B]r = 45[/B]. This is Train 1's Speed.
Train 2's speed = 3(r + 30).
Plugging r = 45 into this, we get 3(45 + 30).
3(75)
[B]225[/B]

A train traveled at 66km an hour for four hours. Find the distance traveled

A train traveled at 66km an hour for four hours. Find the distance traveled
Distance = Rate * Time
Distance = 66km/hr * 4 hours
Distance = [B]264 miles[/B]

A turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtle

A turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtle travels at a constant speed of 20 feet per minute for the entire 100 feet. The rabbit travels at a constant speed of 40 feet per minute for the first 50 feet, stops for 3 minutes, and then continuous at a constant speed of 40 feet per minute for the last 50 feet.
(i) Determine which animal won the race.
(ii). By how much time the animal won the race.
(iii) Explain one life lesson from the race.
We know the distance formula is:
d = rt
For the turtle, he has a rate (r) of 20 feet / minute and distance (d) of 100. We want to solve for time:
[URL='https://www.mathcelebrity.com/drt.php?d=+100&r=+20&t=&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 5
The rabbit has 3 parts of the race:
Rabbit Part 1: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Rabbit Part 2: The rabbit stops for 3 minutes (t = 3)
Rabbit Part 3: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Total time for the rabbit from the 3 parts is (t) = 1.25 + 3 + 1.25
Total time for the rabbit from the 3 parts is (t) = 5.5
[LIST]
[*](i) The [B]turtle won[/B] the race because he took more time to finish and they both started at the same time
[*](ii) We subtract the turtles time from the rabbit's time: 5.5 - 5 = [B]0.5 minutes which is also 30 seconds[/B]
[*](iii) [B]Slow and Steady wins the race[/B]
[/LIST]

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and walked back to the starting point along the same path. She walks 4mph on level ground, 3 mph uphill, and 6 mph downhill. Find the distance she walked.
Hint: Think about d = rt, which means that t = d/r. Think about each section of her walk, what is the distance and the rate. You know that the total time is 5 hours, so you know the sum of the times from each section must be 5.
Let Level distance = L and hill distance = H. Add the times it took for each section of the walk:
L/4 + H /3 + H/6 + L/4 = 5
The LCD of this is 12 from our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=6&pl=LCM']LCD Calculator[/URL]
[U]Multiply each side through by our LCD of 12[/U]
3L + 4H + 2H + 3L = 60
[U]Combine like terms:[/U]
6L + 6H = 60
[U]Divide each side by 3:[/U]
2L + 2H = 20
The woman walked [B]20 miles[/B]

A yardstick casts a shadow of 8 inches. At the same time, a tree casts a shadow of 52 feet. How tall

A yardstick casts a shadow of 8 inches. At the same time, a tree casts a shadow of 52 feet. How tall is the tree?
Setup a proportion of height to shadow distance where h is the height of the tree:
36/8 = h/52
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=36&num2=h&den1=8&den2=52&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get:
h = [B]234 feet[/B]

Alex rode his bike to school at a speed of 12 mph. He then walked home at a speed of 5 mph. What was

Alex rode his bike to school at a speed of 12 mph. He then walked home at a speed of 5 mph. What was Alex's average speed for his trip to school and back?
Say the distance was 1 mile from school to home
D = rt
To school
1 = 12t
t = 1/12
From school:
1 = 5t
t = 1/5
1/2(1/12 + 1/5)
[URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F24&frac2=1%2F10&pl=Add']1/24 + 1/10[/URL] = 17/120
120 = Average speed * 17
Average speed = 120/17 = [B]7.06 mph[/B]

An airplane flies at 250 mph. How far will it travel in 5 h at that rate of speed?

An airplane flies at 250 mph. How far will it travel in 5 h at that rate of speed?
Distance = Rate x Time
Distance = 250mph x 5h
Distance = [B]1,250 miles[/B]

an earthworm moves at distance of 45cm in 90 seconds what is the speed

an earthworm moves at distance of 45cm in 90 seconds what is the speed
Using our [URL='https://www.mathcelebrity.com/drt.php?d=45&r=+&t=90&pl=Calculate+the+missing+Item+from+D%3DRT']distance, rate, time calculator[/URL], we have:
Rate = [B]1/2cm or 0.5cm per second[/B]

An international long distance phone call costs $0.79 per minute. How much will a 22 minute call cos

An international long distance phone call costs $0.79 per minute. How much will a 22 minute call cost?
[U]Calculate total cost:[/U]
Total cost = Cost per minute * number of minutes
Total cost = $0.79 * 22
Total cost = [B]$17.38[/B]

Andrea has 6 hours to spend training for an upcoming race. She completes her training by running ful

Andrea has 6 hours to spend training for an upcoming race. She completes her training by running full speed the distance of the race and walking back the same distance to cool down. If she runs at a speed of 7mph and walks back at a speed of 3mph, how long should she plan to spend walking back?
Let the distance be d.
Running full speed one way, 7d
Walking back the opposite way, 3d
And we know 7d + 3d = 6 hours
10d = 6 hours
d =3/5 hour

Andrea has one hour to spend training for an upcoming race she completes her training by running ful

Andrea has one hour to spend training for an upcoming race she completes her training by running full speed in the distance of the race and walking back the same distance to cool down if she runs at a speed of 9 mph and walks back at a speed of 3 mph how long should she plan on spending to walk back
Let r = running time. Let w = walking time
We're given two equations
[LIST=1]
[*]r + w = 1
[*]9r = 3w
[/LIST]
Rearrange equation (1) by subtract r from each side:
[LIST=1]
[*]w = 1 - r
[*]9r = 3w
[/LIST]
Now substitute equation (1) into equation (2):
9r = 3(1 - r)
9r = 3 - 3r
To solve for r, [URL='https://www.mathcelebrity.com/1unk.php?num=9r%3D3-3r&pl=Solve']we type this equation into our search engine[/URL] and we get:
r = 0.25
Plug this into modified equation (1) to solve for w, and we get:
w = 1. 0.25
[B]w = 0.75[/B]

At 2:18 Pm, a parachutist is 4900 feet above the ground. At 2:32 pm, the parachutist is 2100 above t

At 2:18 Pm, a parachutist is 4900 feet above the ground. At 2:32 pm, the parachutist is 2100 above the ground. Find the average rate of change in feet per minute
Average Rate of Change = Change in Distance / Change in time
Average Rate of Change = (4900 - 2100) / (2:32 - 2:18)
Average Rate of Change = 2800 / 14
Average Rate of Change = [B]200 feet per minute[/B]

Beth made a trip to the train station and back. On the trip there she traveled 45 km/h and on the re

Beth made a trip to the train station and back. On the trip there she traveled 45 km/h and on the return trip she went 30 km/h. How long did the trip there take if the return trip took six hours?
We use the distance formula: D = rt where D = distance, r = rate, and t = time.
Start with the return trip:
D = 45(6)
D = 270
The initial trip is:
270= 30t
Divide each side by 30
[B]t = 9 hours[/B]

Chris walks 12 blocks north and then 16 blocks East. How far is his home from the park

Chris walks 12 blocks north and then 16 blocks East. How far is his home from the park
We've got a right triangle. If we divide 12 and 16 by 4, we get:
12/4 = 3
16/4 = 4
Since the hypotenuse is the distance from the home to the park, we have a classic 3-4-5 right triangle.
So our hypotenuse is 5*4 = [B]20[/B]

Connor runs 2 mi more each day than David. The sum of the distances they run each week is 56 mi. How

Connor runs 2 mi more each day than David. The sum of the distances they run each week is 56 mi. How far does David run each day?
Let Connor's distance be c
Let David's distance be d
We're given two equations:
[LIST=1]
[*]c = d + 2
[*]7(c + d) = 56
[/LIST]
Simplifying equation 2 by dividing each side by 7, we get:
[LIST=1]
[*]c = d + 2
[*]c + d = 8
[/LIST]
Substitute equation (1) into equation (2) for c
d + 2 + d = 8
To solve for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=d%2B2%2Bd%3D8&pl=Solve']type this equation into our calculation engine[/URL] and we get:
d = [B]3[/B]

David roller skates for 3 1/3 hours with a constant speed of 24 km/h and then for another 1 hour 10

David roller skates for 3 1/3 hours with a constant speed of 24 km/h and then for another 1 hour 10 minutes with constant speed of 12 km/h. What distance did he go?
Distance = Rate x Time
[U]Part 1 of his trip:[/U]
D1 = R1 x T1
D1 = 3 & 1/3 hours * 24 km/h
D1 = 80 km
[U]Part 2 of his trip:[/U]
D2 = R2 x T2
D2 = 1 & 1/6 hours * 12 km/h (Note, 10 minutes = 1/6 of an hour)
D2 = 14 km
[U]Calculate Total Distance (D)[/U]
D = D1 + D2
D = 80 + 14
D = [B]94 km[/B]

Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then ad

Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then added a mile to his run each day. By the end of the week (7 days), he had run a total of 70 miles. How many miles did Dennis run the first day?
Setup distance ran for the 7 days:
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[*]n + 4
[*]n + 5
[*]n + 6
[/LIST]
Add them all up:
7n + 21 = 70
Solve for [I]n[/I] in the equation 7n + 21 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 21 and 70. To do that, we subtract 21 from both sides
7n + 21 - 21 = 70 - 21
[SIZE=5][B]Step 2: Cancel 21 on the left side:[/B][/SIZE]
7n = 49
[SIZE=5][B]Step 3: Divide each side of the equation by 7[/B][/SIZE]
7n/7 = 49/7
n =[B] 7
[URL='https://www.mathcelebrity.com/1unk.php?num=7n%2B21%3D70&pl=Solve']Source[/URL][/B]

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

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

distance between -2 and 9 on the number line

distance between -2 and 9 on the number line
Distance on the number line is the absolute value of the difference:
D = |9 - -2|
D = |11|
D = [B]11[/B]

Distance Rate and Time

Solves for distance, rate, or time in the equation d=rt based on 2 of the 3 variables being known.

Each unit on a map of a forest represents 1 mile. To the nearest tenth of a mile, what is the distan

Each unit on a map of a forest represents 1 mile. To the nearest tenth of a mile, what is the distance from a ranger station at (1, 2) on the map to a river crossing at (2, 4) ?
We use our 2 point calculator and we get a distance of 2.2361.
Since each unit represents 1 mile, we have:
2.2361 units * 1 mile per unit = [B]2.2361 miles[/B]

Eric is taking a trip of 245 miles. If he has traveled x miles, represent the remainder of the trip

Eric is taking a trip of 245 miles. If he has traveled x miles, represent the remainder of the trip in terms of x.
Remaining distance = [B]245 - x[/B]

Falling Object

Calculates any of the 3 items in the falling object formula, distance (s), acceleration (a), and time (t).

Find the distance between the points (10,7) and (6,10)

Find the distance between the points (10,7) and (6,10).
[URL='https://www.mathcelebrity.com/slope.php?xone=10&yone=7&slope=+2%2F5&xtwo=6&ytwo=10&pl=You+entered+2+points']Using our two-points calculator[/URL], we get a distance of [B]5[/B].

George and William both ran a one mile race. William won the race with a time of 4 minutes and 30 se

George and William both ran a one mile race. William won the race with a time of 4 minutes and 30 seconds. If George was 480 feet behind William when the race finished, how long did it take George to run the entire mile? (George continued to run at the same pace.)
When the race was done, George completed:
5280 feet in a mile - 480 feet = 4800 feet
set up a proportion of distance traveled to time where n is the time needed to run the mile
4800/4.5 = 5280/n
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=4800&num2=5280&den1=4.5&den2=n&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get:
n = 4.95
5280/4800 = 1.1
Setup another proportion with the 1.1 factor of distance to time:
4800 * 1.1/4.5 * 1.1 = 5280/4.95
4.95 = 4 minutes and .95*60 seconds
4.95 = [B]4 minutes and 57 seconds[/B]

Gigi’s family left their house and drove 14 miles south to a gas station and then 48 miles east to a

Gigi’s family left their house and drove 14 miles south to a gas station and then 48 miles east to a water park. How much shorter would their trip to the water park have been if they hadn’t stopped at the gas station and had driven along the diagonal path instead?
[IMG]https://mathcelebrity.com/community/data/attachments/0/pythag-diagonal.jpg[/IMG]
Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=14&side2input=48&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we see the diagonal route would be:
50 miles
The original trip distance was:
Original Trip Distance = 14 + 48
Original Trip Distance = 62 miles
Diagonal Trip was 50 miles, so the difference is:
Difference = Original Trip Distance - Diagonal Distance
Difference = 62 - 50
Difference = [B]12 miles[/B]

Gravitational Force

Using Sir Isaac Newtons Law of Gravitational Force, this calculator determines the force between two objects with mass in kilograms at a distance apart in meters using the constant of gravity.

Guadalupe left the restaurant traveling 12 mph. Then, 3 hours later, Lauren left traveling the same

Guadalupe left the restaurant traveling 12 mph. Then, 3 hours later, Lauren left traveling the same direction at 24 mph. How long will Lauren travel before catching up with Guadalupe?
Distance = Rate x Time
Guadulupe will meet Lauren at the following distance:
12t = 24(t - 3)
12t = 24t - 72
[URL='https://www.mathcelebrity.com/1unk.php?num=12t%3D24t-72&pl=Solve']Typing that equation into our search engine[/URL], we get:
t = 6

He charges $1.50 per delivery and then $2 per km he has to drive to get from his kitchen to the deli

He charges $1.50 per delivery and then $2 per km he has to drive to get from his kitchen to the delivery address. Write an equation that can be used to calculate the delivery price and the distance between the kitchen and the delivery address. Use your equation to calculate the total cost to deliver to someone 2.4km away
Let k be the number of kilometers between the kitchen and delivery address. Our Delivery equation D(k) is:
[B]D(k) = 2k + 1.50[/B]
The problem wants to know D(2.4):
D(2.4) = 2(2.4) + 1.50
D(2.4) = 4.8 + 1.50
D(2.4) = [B]$6.30[/B]

Hong is riding his bicycle. He rides for 22.5 kilometers at a speed of 9 kilometers per hour. For ho

Hong is riding his bicycle. He rides for 22.5 kilometers at a speed of 9 kilometers per hour. For how many hours does he ride?
Distance = Rate * Time
The problem asks for time.
[URL='https://www.mathcelebrity.com/drt.php?d=+22.5&r=+9&t=&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator[/URL], we get:
t = [B]2.5 hours[/B]

If a soccer ball was kicked at a distance of n decimeters, how many meters did it travel?

If a soccer ball was kicked at a distance of n decimeters, how many meters did it travel?
Meters = Decimeters/10
Meters = [B]n/10[/B]

if a train travels at 80 mph for 15 mins, what is the distance traveled?

if a train travels at 80 mph for 15 mins, what is the distance traveled?
Let d = distance, r = rate, and t = time, we have the distance equation:
D = rt
Plugging in our values for r and t, we have:
D = 80mph * 15 min
Remember our speed is in miles per hour, so 15 min equal 1/4 of an hour
D = 80mph * 1/4
D = [B]20 miles[/B]

If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and dis

If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and distance?
Divide each side by S to isolate T:
D/S = S x T/S
Cancel the S's on the right side:
[B]T = D/S[/B]

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find th

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find the time taken by aeroplane to cover 1200m initially.
We know from the distance formula (d) using rate (r) and time (t) that:
d = rt
Regular speed:
1200 = rt
Divide each side by t, we get:
r = 1200/t
Reduced speed. 20 minutes = 60/20 = 1/3 of an hour. So we multiply 1,200 by 3
3600 = (r - 40)(t + 1/3)
If we multiply 3 by (t + 1/3), we get:
3t + 1
So we have:
3600 = (r - 40)(3t + 1)
Substitute r = 1200/t into the reduced speed equation:
3600 = (1200/t - 40)(3t + 1)
Multiply through and we get:
3600 = 3600 - 120t + 1200/t - 40
Subtract 3,600 from each side
3600 - 3600 = 3600 - 3600 - 120t + 1200/t - 40
The 3600's cancel, so we get:
- 120t + 1200/t - 40 = 0
Multiply each side by t:
-120t^2 - 40t + 1200 = 0
We've got a quadratic equation. To solve for t, [URL='https://www.mathcelebrity.com/quadratic.php?num=-120t%5E2-40t%2B1200%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type this in our search engine[/URL] and we get:
t = -10/3 or t = 3. Since time [I]cannot[/I] be negative, our final answer is:
[B]t = 3[/B]

Janet drove 395 kilometers and the trip took 5 hours. How fast was Janet traveling?

Janet drove 395 kilometers and the trip took 5 hours. How fast was Janet traveling?
Distance = Rate * Time
We're given D = 395 and t = 5
We want Rate. We divide each side of the equation by time:
Distance / Time = Rate * Time / Time
Cancel the Time's on each side and we get:
Rate = Distance / Time
Plugging our numbers in, we get:
Rate = 395/5
Rate = [B]79 kilometers[/B]

Jenny threw the javelin 4 metres further than Angus but 5 metres less than Cameron. if the combined

Jenny threw the javelin 4 metres further than Angus but 5 metres less than Cameron. if the combined distance thrown by the 3 friends is 124 metres, how far did Angus throw the javelin?
Assumptions and givens:
[LIST]
[*]Let a be the distance Angus threw the javelin
[*]Let c be the distance Cameron threw the javelin
[*]Let j be the distance Jenny threw the javelin
[/LIST]
We're given 3 equations:
[LIST=1]
[*]j = a + 4
[*]j = c - 5
[*]a + c + j = 124
[/LIST]
Since j is the common variable in all 3 equations, let's rearrange equation (1) and equation (2) in terms of j as the dependent variable:
[LIST=1]
[*]a = j - 4
[*]c = j + 5
[*]a + c + j = 124
[/LIST]
Now substitute equation (1) and equation (2) into equation (3) for a and c:
j - 4 + j + 5 + j = 124
To solve this equation for j, we [URL='https://www.mathcelebrity.com/1unk.php?num=j-4%2Bj%2B5%2Bj%3D124&pl=Solve']type it in our math engine[/URL] and we get:
j = 41
The question asks how far Angus (a) threw the javelin. Since we have Jenny's distance j = 41 and equation (1) has j and a together, let's substitute j = 41 into equation (1):
a = 41 - 4
a = [B]37 meters[/B]

Jeremy ran 27 laps on a track that was 1/8 mile long. Jimmy ran 15 laps on a track that as 1/4 mile

Jeremy ran 27 laps on a track that was 1/8 mile long. Jimmy ran 15 laps on a track that as 1/4 mile long. who ran farther
[U]Calculate Jeremy's distance:[/U]
Distance = Laps * Track length
Jeremy distance = 27 * 1/8
Jeremy distance = 27/8
[U]Calculate Jimmy's distance:[/U]
Distance = Laps * Track length
Jeremy distance = 15* 1/4
Jeremy distance = 15/4
[COLOR=#000000]Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=27/8&frac2=15/4&pl=Compare']fraction comparison calculator[/URL], we see that [B]Jimmy [/B]ran farther[/COLOR]

Johnny Rocket can run 300 meters in 90 seconds. If his speed remains constant, how far could he ru

Johnny Rocket can run 300 meters in 90 seconds. If his speed remains constant, how far could he run in 500 seconds? Round to one decimal place.
Set up the distance equation:
Distance = Rate * Time
300 = 90r
Solving this equation for r, we [URL='https://www.mathcelebrity.com/1unk.php?num=300%3D90r&pl=Solve']type it in our search engine[/URL] and we get:
r = 3.333
For 500 seconds, we set up our distance equation again:
Distance = 500 * 3.333333
Distance = [B]1666.7 meters[/B]

Kevin ran 4 miles more than Steve ran. The sum of their distances is 26 miles. How far did Steve run

Kevin ran 4 miles more than Steve ran. The sum of their distances is 26 miles. How far did Steve run? The domain of the solution is:
Let k be Kevin's miles ran
Let s be Steve's miles ran
We have 2 given equtaions:
[LIST=1]
[*]k = s + 4
[*]k + s = 26
[/LIST]
Substitute (1) into (2)
(s + 4) + s = 26
2s + 4 = 26
Plug this into our [URL='http://www.mathcelebrity.com/1unk.php?num=2s%2B4%3D26&pl=Solve']equation calculator[/URL] and we get s = 11

last week, bill drove 252 miles. This week, he drove m miles. Using m , write an expression for the

last week, bill drove 252 miles. This week, he drove m miles. Using m, write an expression for the total number of miles he drove in the two weeks
We add the distance driven:
[B]252 + m[/B]

Lena purchased a prepaid phone card for $15. Long distance calls cost 24 cents a minute using this

Lena purchased a prepaid phone card for $15. Long distance calls cost 24 cents a minute using this card. Lena used her card only once to make a long distance call. If the remaining credit on her card is $4.92, how many minutes did her call last?
[U]Figure out how many minutes Lena used:[/U]
Lena spent $15 - $4.92 = $10.08.
[U]Now determine the amount of minutes[/U]
$10.08/0.24 cents per minute = [B]42 minutes[/B]

Let A = (-4,5) and B = (1,3) Find the distance from A to B

Let A = (-4,5) and B = (1,3) Find the distance from A to B
Using our [URL='https://www.mathcelebrity.com/slope.php?xone=-4&yone=5&slope=+&xtwo=1&ytwo=3&bvalue=+&pl=You+entered+2+points']distance between two points calculator[/URL], we get:
[B]5.3852[/B]

Linda can run about 6 yards in one second. About how far can she run in 12 seconds?

Linda can run about 6 yards in one second. About how far can she run in 12 seconds?
Distance = Rate * Time
Distance = 6 yds/ second * 12 seconds
Distance = [B]72 yards[/B]

Line Equation-Slope-Distance-Midpoint-Y intercept

Enter 2 points, and this calculates the following:

* Slope of the line (rise over run) and the line equation y = mx + b that joins the 2 points

* Midpoint of the two points

* Distance between the 2 points

* 2 remaining angles of the rignt triangle formed by the 2 points

* y intercept of the line equation

* Point-Slope Form

* Parametric Equations and Symmetric Equations

Or, if you are given a point on a line and the slope of the line including that point, this calculates the equation of that line and the y intercept of that line equation, and point-slope form.

Also allows for the entry of m and b to form the line equation

* Slope of the line (rise over run) and the line equation y = mx + b that joins the 2 points

* Midpoint of the two points

* Distance between the 2 points

* 2 remaining angles of the rignt triangle formed by the 2 points

* y intercept of the line equation

* Point-Slope Form

* Parametric Equations and Symmetric Equations

Or, if you are given a point on a line and the slope of the line including that point, this calculates the equation of that line and the y intercept of that line equation, and point-slope form.

Also allows for the entry of m and b to form the line equation

Luke drove for n hours at 55 miles per hour. Luke's mother drove for n hours at a speed of 60 miles

Luke drove for n hours at 55 miles per hour. Luke's mother drove for n hours at a speed of 60 miles per hour. How much farther than Luke did his mother drive?
Distance = Rate * Time
[LIST]
[*]Luke drove: 55n
[*]Mom drove 60n
[/LIST]
Distance difference = 60n - 55n = [B]5n[/B]

Maria called her sister long distance on Wednesday. The first 5 minutes cost $3, and each minute aft

Maria called her sister long distance on Wednesday. The first 5 minutes cost $3, and each minute after that cost $0.25. How much did it cost if they talked for 15 minutes?
First 5 minutes: $3
If they talked 15 minutes, the additional charge past 5 minutes is:
0.25 * (15 - 5)
0.25 * 10 minutes = $2.5
We add this to the first 5 minutes:
$3 + $2.5 = [B]$5.50[/B]

Maria leaves her house and runs west for 6 m miles. She then turns North and runs 5 miles. Maria the

Maria leaves her house and runs west for 6 miles. She then turns North and runs 5 miles. Maria then travels east for 7 miles and then south for 5 miles. How far is Maria from her house now?
Maria traveled the same distance north and south of 5 miles. These cancel each other out.
Her 7 mile eastern trip compared to the 6 mile west trip represents a net difference of [B]1 mile[/B]

Midpoint formula

Midpoint formula
Given two points (x1, y1) and (x2, y2), the midpoint is found as the average distance between the 2 points:
[LIST]
[*]x value is: (x1 + x2)/2
[*]y value is: (y1 + y2)/2
[/LIST]
So our midpoint is:
((x1 + x2)/2, (y1 + y2)/2)

On a map, every 5 cm represents 250 kilometres. What distance would be represented by a 3 cm line?

On a map, every 5 cm represents 250 kilometres. What distance would be represented by a 3 cm line?
We set up a proportion of map cm distance to kilometers where k is the kilometers represented by a 3cm line
5/250 = 3/k
To solve this proportion for k, we [URL='https://www.mathcelebrity.com/prop.php?num1=5&num2=3&den1=250&den2=k&propsign=%3D&pl=Calculate+missing+proportion+value']type it in our search engine[/URL] and we get:
k = [B]150[/B]

Opposite Direction Distance

Word Problem calculator to measure distance between 2 people moving in opposite directions with rate and time solved for as well

please answer my second word problem

Distance = Rate x Time
6.4 meters = 4 meters/minute * t
Divide each side by 4
[B]t = 1.6 minutes[/B]

please answer this word problem

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

please solve the fifth word problem

Karen purchased a prepaid phone card for
$20
. Long distance calls cost
11
cents a minute using this card. Karen used her card only once to make a long distance call. If the remaining credit on her card is
$17.47
, how many minutes did her call last?

Point and a Line

Enter any line equation and a 2 dimensional point. The calculator will figure out if the point you entered lies on the line equation you entered. If the point does not lie on the line, the distance between the point and line will be calculated.

Rachel runs 2 miles during each track practice. Write an equation that shows the relationship betwe

Rachel runs 2 miles during each track practice. Write an equation that shows the relationship between the practices p and the distance d.
Distance equals rate * practicdes, so we have:
[B]d = 2p[/B]

Rebound Ratio

Calculates a total downward distance traveled given an initial height of a drop and a rebound ratio percentage

Salma purchased a prepaid phone card for 30. Long distance calls cost 9 cents a minute using this ca

Salma purchased a prepaid phone card for 30. Long distance calls cost 9 cents a minute using this card. Salma used her card only once to make a long distance call. If the remaining credit on her card is 28.38, how many minutes did her call last?
[U]Set up the equation where m is the number of minutes used:[/U]
0.09m = 30 - 28.38
0.09m = 1.62
[U]Divide each side by 0.09[/U]
[B]m = 18[/B]

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from the school?
Sam walked at a right angle. His distance from home to school is the hypotenuse.
Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=8&side2input=10&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we get:
[B]12.806 blocks[/B]

Set Notation

Given two number sets A and B, this determines the following:

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J_{σ}(A,B)

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is runn

Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is running at 5 feet per second and Claire is running at 3 feet per second.
i. After how many seconds will Sophie catch Claire?
ii. If the race is 500 feet, who wins?
i.
Sophie's distance formula is given as D = 5s
Claire's distance formula is given as D = 3s + 100
Set them equal to each other
5s = 3s + 100
Subtract 3s from both sides:
2s = 100
Divide each side by 2
[B]s = 50[/B]
ii. [B]Sophie since after 50 seconds, she takes the lead and never gives it back.[/B]

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

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

Stopping-Braking Distance for a Car

Calculates the estimated stopping distance of a vehicle given a speed in miles per hour (mph)

String Comparison Algorithms

Given two strings A and B, this calculates the following items:

1) Similar Text Pair Ranking Score

2) Levenshtein (Edit Distance).

1) Similar Text Pair Ranking Score

2) Levenshtein (Edit Distance).

Stuart traveled n miles at a speed of 72 miles per hour. How many seconds did it take Stuart to trav

Stuart traveled n miles at a speed of 72 miles per hour. How many seconds did it take Stuart to travel the n miles?
Distance = Rate * Time
Time = Distance/Rate
Time = n/72 hours
3600 seconds per hour so we have:
3600n/72
[B]50n[/B]

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed wit

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If X = average distance in feet for 49 fly balls, then X ~ _______(_______,_______)

b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80^{th} percentile of the distribution of the average of 49 fly balls
a. N(250, 50/sqrt(49)) = [B]0.42074[/B]
b. Calculate Z-score and probability = 0.08 shown [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+240&mean=+250&stdev=+7.14&n=+1&pl=P%28X+%3C+Z%29']here[/URL]
c. Inverse of normal distribution(0.8) = 0.8416. Use NORMSINV(0.8) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.8&pl=Calculate+Critical+Z+Value']calculator[/URL]
Using the Z-score formula, we have
0.8416 = (x - 250)/50
x = [B]292.08[/B]

b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed wit

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and
a standard deviation of 50 feet.
a. If X = distance in feet for a fly ball, then X ~
b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.
a. [B]N(250, 50/sqrt(1))[/B]
b. Calculate [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+220&mean=250&stdev=50&n=+1&pl=P%28X+%3C+Z%29']z-score[/URL]
Z = -0.6 and P(Z < -0.6) = [B]0.274253[/B]
c. Inverse of normal distribution(0.8) = 0.8416 using NORMSINV(0.8) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.8&pl=Calculate+Critical+Z+Value']calculator[/URL]
Z-score formula: 0.8416 = (x - 250)/50

x = [B]292.08[/B]

x = [B]292.08[/B]

the absolute value of a number is its _____ from 0

the absolute value of a number is its _____ from 0
The answer is [B]distance[/B].
As an example: 2 and -2 are 2 units away from 0.

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

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

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base line about 40 feet behind third base. How far would the outfielder have to throw the ball to first base?
We have a right triangle. From home base to third base is 90 feet. We add another 40 feet to the outfielder behind third base to get: 90 + 40 = 130
The distance from home to first is 90 feet.
Our hypotenuse is the distance from the outfielder to first base.
[URL='https://www.mathcelebrity.com/pythag.php?side1input=130&side2input=90&hypinput=&pl=Solve+Missing+Side']Using our Pythagorean theorem calculator[/URL], we get:
d = [B]158.11 feet[/B]

The distance between X and 8 is less than 14

Distance implies the positive difference between 2 points. Therefore, we use absolute value:
|x - 8| < 14
Note, we use less than since 14 is not included.

The distance traveled in t hours by a car traveling at 65 miles per hour

The distance traveled in t hours by a car traveling at 65 miles per hour.
Distance = Rate * Time
Distance = 65 mph * t hours
Distance = [B]65t[/B]

The famous Concorde jet travelled at a speed of 2000km/h for two and a half hours. Do you think it c

The famous Concorde jet travelled at a speed of 2000km/h for two and a half hours. Do you think it could make it to its destination which is 5500km away on time
Calculate the total distance traveled @ 2000km/h for 2.5 hours:
d = rt
d = 2000 * 2.5
d = 5,000 km
The answer is [B]no, it cannot make the destination[/B].

The marianas trench is 10415m below the sea level. Directly above it a helicopter is hovering 3200m

The marianas trench is 10415m below the sea level. Directly above it a helicopter is hovering 3200m above sea level. How far is the helicopter from the trench?
Below sea level is negative:
-10415
Above sea level is positive:
+3200
The distance is found by:
+3200 - -10415
+3200 + 10415
[B]13,615[/B]

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company called Rachel and offered her long distance service for 1/2 cent per minute, but will charge a $46 monthly fee. How many minutes per month must Rachel talk on the phone to make the discount a better deal?
Minutes Rachel talks = m
Current plan cost = 0.12m
New plan cost = 0.005m + 46
Set new plan equal to current plan:
0.005m + 46 = 0.12m
Solve for [I]m[/I] in the equation 0.005m + 46 = 0.12m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.005m and 0.12m. To do that, we subtract 0.12m from both sides
0.005m + 46 - 0.12m = 0.12m - 0.12m
[SIZE=5][B]Step 2: Cancel 0.12m on the right side:[/B][/SIZE]
-0.115m + 46 = 0
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 46 and 0. To do that, we subtract 46 from both sides
-0.115m + 46 - 46 = 0 - 46
[SIZE=5][B]Step 4: Cancel 46 on the left side:[/B][/SIZE]
-0.115m = -46
[SIZE=5][B]Step 5: Divide each side of the equation by -0.115[/B][/SIZE]
-0.115m/-0.115 = -46/-0.115
m = [B]400
She must talk over 400 minutes for the new plan to be a better deal
[URL='https://www.mathcelebrity.com/1unk.php?num=0.005m%2B46%3D0.12m&pl=Solve']Source[/URL][/B]

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be equal to 3 miles?
Set up a proportion of scale to actual distance
1/2 / 3/4 = x/3
4/3 = x/3
Cross multiply:
3x = 12
Divide each side by 3:
3x/3 = 12/3
x = [B]4 (1/2 inch sections) or 2 inches[/B]

The world record for the mile in the year 1865 was held by Richard Webster of England when he comple

The world record for the mile in the year 1865 was held by Richard Webster of England when he completed a mile in 4 minutes and 36.5 seconds. The world record in 1999 was set by Hicham El Guerrouj when he ran a mile in 3 minutes and 43.13 seconds.
If both men ran the mile together, how many feet behind would Richard Webster be when Hichem El Guerrouj crossed the finish line?
Change times to seconds:
[LIST]
[*]4 minutes and 36.5 seconds = 4*60 + 36.5 = 240 + 36.5 = 276.5 seconds
[*]3 minute and 43.13 seconds = 3*60 + 43.13 = 180 + 43.13 = 223.13 seconds
[/LIST]
Now, find the distance Richard Webster travelled in 3 minutes and 43.13 seconds which is when Hiram El Guerrouj crossed the finish line.
1 mile = 5280 feet:
Set up a proportion of distance in feet to seconds where n is the distance Richard Webster travelled
5280/276.5 = n/223.13
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=5280&num2=n&den1=276.5&den2=223.13&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get:
n = 4260.85 feet
Distance difference = 5280 - 4260.85 = [B]1019.15 feet[/B]

There is an escalator that is 1090.3 feet long and drops a vertical distance of 193.4 feet. What is

There is an escalator that is 1090.3 feet long and drops a vertical distance of 193.4 feet. What is its angle of depression?
The sin of the angle A is the length of the opposite side / hypotenuse.
sin(A) = Opposite / Hypotenuse
sin(A) = 193.4 / 1090/3
sin(A) = 0.1774
[URL='https://www.mathcelebrity.com/anglebasic.php?entry=0.1774&pl=arcsin']We want the arcsin(0.1774)[/URL].
[B]A = 10.1284[/B]

Tristan is building a slide for his kids. The ladder is 6 feet tall and the slide is 10 feet long. W

Tristan is building a slide for his kids. The ladder is 6 feet tall and the slide is 10 feet long. What is the distance between the ladder and the bottom of the slide?
The answer is 8.
We have a 3-4-5 triangle. But it's scaled by 2.
3 * 2 = 6
5 * 2 = 10 (hypotenuse-slide)
4 * 2 = [B]8[/B]

Using a number line how far is - 2 from 6

Using a number line how far is - 2 from 6
We use [URL='https://www.mathcelebrity.com/mptnline.php?ept1=-2&empt=+&ept2=6&pl=Calculate+missing+Number+Line+item']our number line calculator[/URL] and we get:
Distance is [B]8[/B]

Vectors

Given 2 vectors A and B, this calculates:

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj_{B}A and and the vector component of A orthogonal to B → A - proj_{B}A

Also calculates the horizontal component and vertical component of a 2-D vector.

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj

Also calculates the horizontal component and vertical component of a 2-D vector.

Walking Distance (Pedometer)

Given a number of steps and a distance per stride in feet, this calculator will determine how far you walk in other linear measurements.

When a dog noticed a fox, they were 60 meters apart. The dog immediately started to chase the fox at

When a dog noticed a fox, they were 60 meters apart. The dog immediately started to chase the fox at a speed of 750 meters per minute. The fox started to run away at a speed of 720 meters per minute. How soon will the dog catch the fox?
The dog sits a position p.
Distance = Rate x Time
The dogs distance in minutes is D = 720t
The fox sits at position p + 60
Distance = Rate x Time
The fox's distance in minutes is D = 750t - 60 <-- Subtract 60 since the fox is already ahead 60 meters.
We want to know when their distance (location) is the same. So we set both distance equations equal to each other:
720t = 750t - 60
[URL='https://www.mathcelebrity.com/1unk.php?num=720t%3D750t-60&pl=Solve']Using our equation calculator[/URL], we get [B]t = 2[/B].
Let's check our work:
Dog's distance is 720(2) = 1440
Fox's distance is 750(2) - 60 = 1,440

which number is the same distance from 0 on the number line as 4

which number is the same distance from 0 on the number line as 4
We use absolute value for distance.
Since 4 is 4 units right of 0 on the number line, we can also move 4 units left of 0 on the number line and we land on [B]-4[/B]

Work

Solves for any of the 3 variables, Work (W), Force (F) and Distance (d) in the work formula

Yolanda is riding her bicycle. She rides for 5 hours at a speed of 12.5 kilometers per hours. For ho

Yolanda is riding her bicycle. She rides for 5 hours at a speed of 12.5 kilometers per hours. For how many kilometers does she ride?
This is a distance problem, where distance = rate * time. We are given time of 5 hours, at a rate of 12.5km/hour.
Using our [URL='http://www.mathcelebrity.com/drt.php?d=+&r=12.5&t=5&pl=Calculate+the+missing+Item+from+D%3DRT']distance calculator[/URL], we get D = [B]62.5km[/B].