Question

two high speed trains are 480 miles apart and traveling towards each other. they meet in 3 hours. if one trains speed is 5 miles per hour faster than the other, find the speed of each train

Answer 1

We can model the problem as follows: on a number line we put train A, with positive speed
`v_A`, at the origin x=0.

At x=480 we put train B, with negative speed
`v_B`.

So, the equations for the positions of the two trains are

`\begin{cases}x_A = v_At\\x_B=480-v_Bt\end{cases}`

We know that
`v_A=v_B+5`, so we can rewrite the first equation:

`\begin{cases}x_A = (v_B+5)t\\x_B=480-v_Bt\end{cases}`

The two trains meet when they are at the same position:

`x_A=x_B \iff (v_B+5)t=480-v_Bt`

We know that this happens after 3 hours, i.e. when t=3:

`3(v_B+5)=480-3v_B \iff 3v_B+15=480-3v_B \iff 6v_B = 465 \iff v_B=77.5`

And since train A was 5 mph faster, we have

`v_A=77.5+5=82.5`

Answer 2

77.5 mph and 82.5 mph

Explanation:

Let's say the speeds of the trains are x and x + 5.

The combined distance traveled by the trains is 480 miles.

480 = x × 3 + (x + 5) × 3

480 = 3x + 3x + 15

480 = 6x + 15

465 = 6x

x = 77.5

The speeds of the trains are 77.5 mph and 82.5 mph.