Example 3 from Algebra and Trigonometry, 2.3 Models and Applications

Solving an Application Using a Formula

It takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This is a distance problem, so we can use the formula $d = r t,$ where distance equals rate multiplied by time. Note that when rate is given in mi/h, time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution.

First, we identify the known and unknown quantities. Andrew’s morning drive to work takes 30 min, or $\frac{1}{2}$ h at rate $r .$ His drive home takes 40 min, or $\frac{2}{3}$ h, and his speed averages 10 mi/h less than the morning drive. Both trips cover distance $d .$ A table, such as Table 2, is often helpful for keeping track of information in these types of problems.

$d$ $r$ $t$

To Work $d$ $r$ $\frac{1}{2}$

To Home $d$ $r - 10$ $\frac{2}{3}$

Table 2

Write two equations, one for each trip.

$\begin{matrix} d & = & r (\frac{1}{2}) & To work \\ d & = & (r - 10) (\frac{2}{3}) & To home \end{matrix}$
As both equations equal the same distance, we set them equal to each other and solve for r.

$\begin{matrix} r (\frac{1}{2}) & = & (r - 10) (\frac{2}{3}) \\ \frac{1}{2} r & = & \frac{2}{3} r - \frac{20}{3} \\ \frac{1}{2} r - \frac{2}{3} r & = & - \frac{20}{3} \\ - \frac{1}{6} r & = & - \frac{20}{3} \\ r & = & - \frac{20}{3} (−6) \\ r & = & 40 \end{matrix}$
We have solved for the rate of speed to work, 40 mph. Substituting 40 into the rate on the return trip yields 30 mi/h. Now we can answer the question. Substitute the rate back into either equation and solve for d.

$\begin{matrix} d & = & 40 (\frac{1}{2}) \\ = & 20 \end{matrix}$
The distance between home and work is 20 mi.

	$d$	$r$	$t$
To Work	$d$	$r$	$\frac{1}{2}$
To Home	$d$	$r - 10$	$\frac{2}{3}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

On Saturday morning, it took Jennifer 3.6 h to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

Your turn