Chapter 5.8, Problem 21P

To determine

To find how far apart two cities .

Answer to Problem 21P

The distance between two cities is 160 km.

Explanation of Solution

Formula:

Given that the rail line between two cities consists of two segments, one 96 km longer than the other.

Now, A passenger train average $60km/h$ over the shorter segment, $120km/h$ over the longer , and $100km/h$ for the entire trip.

Formula:

  $\text{time=}\frac{\text{distance}}{\text{speed}}$

Let the shorter segment of rail be x then the longer segment is $(x+96)$

Therefore, the total length of the rail line is $\text{(shorter+longer)segment=x+x+96=2x+96}$

First, to determine the time of the shorter segment where distance is x and speed $60km/h$

  $t_s=\frac{x}{60}\mathrm{..............}(i)$

similarly to determine the time of the longer segment where distance is x+96 and speed $120km/h$

  $t_l=\frac{x+96}{120}\mathrm{............}(ii)$

Now, the total time $t_t=\frac{2x+96}{100}\mathrm{............}(iii)$

Thus , the total time is equal to the time of shorter segment with the longer segment time

Now, Add equation ( i ) and equation ( ii ) and compare with equation ( iii)

  $\begin{array}{l}t=t_s+t_l\\ \frac{2x+96}{100}=\frac{x}{60}+\frac{x+96}{120}\end{array}$

Simplify the equation

  $\begin{array}{c}\frac{2x+96}{5}=\frac{x}{3}+\frac{x+96}{6}\\ 6(2x+96)=5(2x+x+96)\\ 12x+576=15x+480\\ 3x=96\\ x=32\end{array}$

Hence, the total distance between the two cities is total distance $\text{2x+96=2(32)+96=160 km}$