Chapter 1.7, Problem 26E

Solution

To calculate the number of hours Bruins basketball team spent driving on local highways.

Bruins basketball team spent $4.5\text{ hours}$ of driving on local highways.

Given:

Total distance of trip = $560\text{ miles}$

Speed on local highways = $45\text{ mph}$

Speed on interstate highways = $55\text{ mph}$

Number of hours spent more on the interstate highways than local highways = $2\text{ hours}$

Formula used: Average speed of the team is determined by calculating the ratio of distance travelled to time taken to cover this distance, i.e.

  $\text{Speed }(v)=\frac{\text{Distance }(d)}{\text{Time }(t)}$

Calculation:

Let the time spent on local highways = $t_l$

Let the time spent on interstate highways = $t_i$

Let the average speed on local highways = $v_l$

Let the average speed on interstate highways = $v_i$

Let total distance = $d_t$

Let the distance of local highways = $d_l$

Let the distance of interstate highways = $d_i$

Now, from the given information, we have:

  $\begin{array}{l}{t}_i=t_l+2\\ v_l=45\text{ mph}\\ v_i=55\text{ mph}\\ {\text{d}}_t=560\text{ miles}\end{array}$

Now, the average speed on local highways is given by:

  $\begin{array}{l}{v}_l=\frac{d_l}{t_l}\\ 45=\frac{d_l}{t_l}\cdots (1)\end{array}$

And, the total distance travelled is given by:

  $d_l+d_i=560\cdots (2)$

And, the average speed on interstate highways is given by:

  $\begin{array}{l}{v}_i=\frac{d_i}{t_i}\\ 55=\frac{560-d_l}{t_l+2}\cdots (3)\end{array}$

From equation (1), we have:

  $d_l=45t_l$

Substituting the value of $d_l$ in equation $(3)$ , we have:

  $\begin{array}{l}55=\frac{560-45t_l}{t_l+2}\\ \Rightarrow 55t_l+110=560-45t_l\\ \Rightarrow 55t_l+45t_l=560-110\\ \Rightarrow 100t_l=450\\ \Rightarrow t_l=\frac{450}{100}\\ \Rightarrow t_l=4.5\text{ hours}\end{array}$

Hence, Bruins basketball team spent $4.5\text{ hours}$ of driving on local highways.