Question

The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least ____ mi/hr/s.

Solution

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Mean Value Theorem\textbf{Mean Value Theorem}: if ff is a continuous function on the closed interval [a,b][a, b], and differentiable on the open interval (a,b)(a, b), then there exists a point cc in (a,b)(a, b) such that:

f(c)=f(b)f(a)ba\begin{gather*} f'(c)=\dfrac{f(b)-f(a)}{b-a} \end{gather*}

We are given that f(0)=0f(0)=0 and f(4.45)=330f(4.45)=330. Therefore,

f(c)=33004.450f(c)=3304.45f(c)74.16 mi/hr/s\begin{gather*} f'(c)=\dfrac{330-0}{4.45-0}\\ f'(c)=\dfrac{330}{4.45}\\ f'(c)\approx74.16 \text{ mi/hr/s} \end{gather*}

At some point during the race, the maximum acceleration of the drag racer is at least 74.16 mi/hr/s.

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