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QuestionTwo buses start 516 miles apart and meet in 4 hours. One is 15mih15 \frac{\mathrm{mi}}{\mathrm{h}} slower. Find their speeds.

Studdy Solution

STEP 1

Assumptions1. The two buses start516 miles apart. . They travel towards each other.
3. One bus travels15 mi/h slower than the other.
4. They meet after4 hours.

STEP 2

Let's denote the speed of the faster bus as xx (mi/h). Therefore, the speed of the slower bus is x−15x -15 (mi/h).

STEP 3

We know that the total distance covered by both buses is516 miles. Since they are traveling towards each other, the total distance covered is the sum of the distances covered by each bus.
Total distance=Distancefaster bus+Distanceslower busTotal\, distance = Distance_{faster\, bus} + Distance_{slower\, bus}

STEP 4

We also know that distance is the product of speed and time. So, we can express the distances covered by each bus in terms of their speeds and the time they traveled.
Total distance=x⋅time+(x−15)⋅timeTotal\, distance = x \cdot time + (x -15) \cdot time

STEP 5

Now, we can plug in the given values for the total distance and the time to get an equation in terms of xx.
516=4x+4(x−15)516 =4x +4(x -15)

STEP 6

implify the equation by distributing the4 on the right side.
516=4x+4x−60516 =4x +4x -60

STEP 7

Combine like terms on the right side of the equation.
516=x−60516 =x -60

STEP 8

Add60 to both sides of the equation to isolate the term with xx on one side.
516+60=8x516 +60 =8x

STEP 9

implify the left side of the equation.
576=8x576 =8x

STEP 10

Divide both sides of the equation by8 to solve for xx.
x=5768x = \frac{576}{8}

STEP 11

Calculate the value of xx.
x=72x =72So, the speed of the faster bus is72 mi/h.

STEP 12

Now, we can find the speed of the slower bus by subtracting15 from the speed of the faster bus.
Speedslower bus=x−15Speed_{slower\, bus} = x -15

STEP 13

Plug in the value for xx to calculate the speed of the slower bus.
Speedslower bus=72−15Speed_{slower\, bus} =72 -15

STEP 14

Calculate the speed of the slower bus.
Speedslower bus=57Speed_{slower\, bus} =57So, the speed of the slower bus is57 mi/h.
The rates of the buses are72 mi/h and57 mi/h.

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