Question

A train travels 110 miles in the same time that a plane covers 528 miles. If the speed of the plane is 10 miles per hour less than 5 times thespeed of the train, find both speeds.

Answer 1

We will have the following:

First, we will have that the expressions of distance for the train and plane are respectively:

[We remember that velocity times time equals distance]

`\begin{cases}t\colon v\cdot t=110mi \\ \\ p\colon(5v-10)\cdot t=528mi\end{cases}`

From this we solve for time "t" in each expression:

`\begin{cases}t=(110mi)/(v) \\ \\ t=(528mi)/(5v-10)\end{cases}`

Now, since the time is the same, we equal both expressions, that is:

`(110mi)/(v)=(528mi)/(5v-10)`

Now, we solve for "v"

`\Rightarrow110(5v-10)=528(v)\Rightarrow550v-1100=528v`
`\Rightarrow22v=1100\Rightarrow v=50`

Now, we determine each vehicle's velocity:

`\begin{cases}t\colon50mi/h \\ \\ p\colon240mi/h\end{cases}`

So, the train is moving at 50 miles per hour and the plane is moving at 240 miles per hour.

We corroborate by determining the time it takes for each to reach the distances stated, that is:

*Train:

`(50mi/h)\cdot t=110mi\Rightarrow t=(11)/(5)h\Rightarrow t=2.2h`

*Plane:

`(240mi/h)\cdot t=528mi\Rightarrow t=(11)/(5)h\Rightarrow t=2.2h`

Thus proving that the velocities are the ones calculated.