Chapter 1.9, Problem 19P

To determine

To calculate:

The time of the train cross each other:

Answer to Problem 19P

The time of trains cross each other after $2.5$ hr.

Explanation of Solution

Given information:

The train towards east: $90$ min/hr

The train towards west: $100$ min/hr

Total distance: $450$ min

Calculation:

Let, the time when the $2$ trains pass to each other is $t$ . Thus, the time taken by the second train is $t-\frac{1}{4}$ :

  

Use the formula of the distance:

  $D=R\times T\mathrm{...}\left(1\right)$

Where,

  $D$ : Distance

  $R$ : Rate

  $T$ : Time

For the train towards east:

  $D=90\cdot t$

For the train towards west:

  $\begin{array}{l}D=100\times \left(t-\frac{1}{4}\right)\\ D=100\left(t-\frac{1}{4}\right)\end{array}$

The total distance covered the two trains is $450$ mi:

  $90t+100\left(t-\frac{1}{4}\right)=450$

Use the distributive property:

  $\begin{array}{l}90t+100t-\frac{100}{4}=450\\ 90t+100t-25=450\\ 190t-25=450\end{array}$

Add to $25$ both sides of the equation:

  $\begin{array}{l}190t-25+25=450+25\\ 190t=475\end{array}$

Divide both the sides by $190$ :

  $\begin{array}{l}\therefore \frac{190}{190}t=\frac{475}{190}\\ \therefore t=2.5\end{array}$

The trains cross each other after $2.5$ hr.

The total distance gives $450$ :

  $\begin{array}{l}\therefore 90\left(2.5\right)+100\left(2.5-\frac{1}{4}\right)=450\\ \therefore 90\left(2.5\right)+250-25=450\\ \therefore 225+225=450\\ \therefore 450=450\end{array}$

The two trains pass each other after at $2$ hr $30$ min or $2.5$ hr.