Chapter 1.2, Problem 14E

(a)

To determine

To express: The distance in terms of the time elapsed.

Answer to Problem 14E

The distance d in terms of time elapsed is, $d=48t$.

Explanation of Solution

Let d be the distance traveled in miles and the t be the time elapsed in hours.

When t = 0, the distance is 0 which can be expressed as (0, 0).

Since he passes the Ann Arbor at 50 minutes, t = 50 minutes or $t=\frac{5}{6}\text{hrs}\text{.}$

The distance traveled is 40 miles. That is, at $t=\frac{5}{6}\text{hrs}\text{.}$, $d=40$ which can be expressed as $\left(\frac{5}{6},40\right)$.

Use the obtained two points and find the slope m as follows.

$\begin{array}{c}m=\frac{40-0}{\frac{5}{6}-0}\\ =40\times \frac{6}{5}\\ =48\end{array}$

Thus, the slope $m=48$.

Use the slope $m=48$ and the point (0, 0) and obtain the equation of distance d in terms of time elapsed t as follows.

$\begin{array}{l}y-y_1=m\left(t-x_1\right)\\ d-0=48\left(t-0\right)\\ d=48t\end{array}$

Thus, the required function is $d=48t$.

(b)

To determine

To sketch: The graph of the distance equation obtained in part (a).

Explanation of Solution

Let x-axis be represented the time in hours and the y-axis be represented the distance in miles.

Given that the distance function of elapsed time is $d=40t$.

Obtain the value of d for various values of t and draw the graph of d as shown below in Figure 1.

From Figure 1, it is observed that function is a linear function.

(c)

To determine

To find: The slope of the graph drawn in part (b); explain the meaning of slope.

Answer to Problem 14E

The slope is 48, which represents the rate of change of the distance in miles.

Explanation of Solution

The equation $d=40t$ is in the form of $y=mx+c$ in which m is slope and c is the y-intercept.

Thus, the slope of the Figure 1 shown in part (b) is 48. It represents that the rate of change of the distance in miles.

Moreover, it is observed that there is a decrease in the distance when time increases and hence it is in direct variation. The distance covered in an hour is 48 miles.