Chapter 3.7, Problem 38PPS

To determine

To calculate: The total time in hours the bus was on the highway and the total time in hours the bus was driving in the city.

Answer to Problem 38PPS

The hours bus was on the highway = $2$ hours.

The hours bus was in the driving city = $5$ hours.

Explanation of Solution

Given information:

The average speed of bus on highway= 65 miles per hour.

The average speed of bus in the city =25 miles per hour distance between art museum and school =375 miles.

Time taken to cover the distance= 7 hours.

Calculation:

Consider the case of Mr. Smith taking a bus trip to art museum. The speed of the bus was 65 miles per hours on the highway and 25 miles per hour in the city.

Let number of hours taken by the bus in the city be $x$ ,

Let number of hours taken by the buson the highway be $y$

Distance travelled by the bus in city,

  $t_1=65x$

Distance travelled by the bus on highway,

  $t_2=25x$

Total distance is given as 375 miles. Hence,

  $65x+25y=375$

I took 7 hours to reach art museum. Hence,

  $x+y=7$

Hence the set of simultaneous equations are:

  $\begin{array}{l}65x+25y=375\\ x+y=7\end{array}$

Solve equations using Cramer's rule as follows.

Coefficient matrix is,

  $\left[\begin{array}{cc}65& 25\\ 1& 1\end{array}\right]$

Determinant of the matrix is.

  $\begin{array}{cc}C& =\left|\begin{array}{cc}65& 25\\ 1& 1\end{array}\right|\\ & =65\times 1-25\times 1\\ & =40\end{array}$

Using Cramer's rule the solution is,

  $\begin{array}{l}x=\frac{\left|\begin{array}{cc}m& b\\ n& h\end{array}\right|}{\left|C\right|}\\ =\frac{\left|37525\right|}{40}\\ =\frac{(375\times 1-25\times 7)}{40}\\ =\frac{200}{40}\\ =5\end{array}$

And,

  $\begin{array}{l}y=\frac{\left|\begin{array}{cc}a& m\\ f& n\end{array}\right|}{\left|C\right|}\\ \frac{=\left|\begin{array}{cc}65& 375\\ 1& 7\end{array}\right|}{40}\\ =\frac{(65\times 7-375)}{40}\\ =\frac{80}{40}\\ =2\end{array}$

Hence the solution is shown below.

  $\begin{array}{l}x=5\\ y=2\end{array}$

Hence the bus travels for 5 hours in city and 2 on the highway.