Preview Activity 1.5.1.
Suppose that you are driving a car on the highway using cruise control so that you drive at a constant speed. The time, \(T\) (measured in hours), it takes to drive 50 miles depends on the constant rate, \(s\) (measured in miles per hour), at which you are driving. Since
\begin{equation*}
\mbox{distance} = \mbox{rate} \cdot \mbox{travel time}\text{,}
\end{equation*}
it follows that
\begin{equation*}
\mbox{travel time} = \frac{\mbox{distance}}{\mbox{rate}}\text{,}
\end{equation*}
and thus we know that \(T(s) = \frac{50}{s}\text{.}\)
For instance, observe that \(T(75) = \frac{50}{75} = \frac{2}{3}\text{,}\) so it takes \(\frac{2}{3}\) of an hour to drive \(50\) miles at \(75\) miles per hour, and the point \((75, \frac{2}{3})\) lies on the graph of \(T(s)\text{.}\)
(a)
Plot the graph of \(T(s) = \frac{50}{s}\) on the axes provided in Figure 1.5.1. Include the point \((75, \frac{2}{3})\text{,}\) and plot and label 2 additional points of your choice. Write a sentence to explain why the shape of the graph makes sense.
(b)
What is the value of \(T(60)\text{?}\) Include units on your answer. Write a sentence to explain the meaning of this value in context.
(c)
Recall the average rate of change of a function \(f\) on an interval \([a,b]\text{,}\) given by \(AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}\text{.}\) Determine the average rate of change of \(T\) on the interval \([60,65]\text{,}\) \(AV_{[60, 65]}\text{.}\) Include units on your answer.
(d)
Is the sign of \(AV_{[60, 65]}\) positive or negative? Why does this sign make sense in light of the graph of \(T(s)\text{?}\) Explain your thinking.
(e)
Write a sentence in everyday language that explains the meaning of the value of \(AV_{[60, 65]}\text{.}\) Include units in your sentence.
