Q. 63 - Calculus

Question

It took Alina half an hour to drive to the grocery store that is 20 miles from her house.

(a) Use the Mean Value Theorem to show that, at some point during her trip, Alina must have been traveling exactly 40 miles per hour.

(b) Why does what you have shown in part (a) make sense in real-world terms?

Step-by-Step Solution

Answer

(a) Yes, Alina was traveling at 40 miles per hour towards the store.

(b) Could Alina have been traveling 40  miles per hour for the whole trip.

Part (a). Step 1. Given Information.

Alina took half an hour to drive to a grocery store 20 miles from the house.

Part (a). Step 2. Mean Value Theorem.

Let s(t) miles be Alina's distance from the grocery store at time t.

Then,

$s\left(0\right)=20;s(0,5)=0$

Therefore, s(t) is continuous and differentiable. Applying Mean Value Theorem, there is sometimes $c\in (0,0.5)$ where,

$$\begin{gathered} s\text{'}\left(c\right)=\frac{f(0.5)-f\left(0\right)}{0.5-0} \\ =\frac{0-20}{0,5} \\ =-40 \end{gathered}$$

Since the sign is negative, it means she is traveling towards the store.

Part (b). Step 1. Real-life example.

Could Alina have been traveling 40  miles per hour for the whole trip.