Question

the horsepower, h(s), required for a racecar to overcome wind resistance is given by the function ( h(s) = 0.003s^2 + 0.07s - 0.027 ), where ( s ) is the speed of the car in miles per hour. what is the average rate of change in horsepower per unit speed if the racecar increases in speed from 80 mph to 100 mph? options: 1.64, 12.2, 0.61, 20.0

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( H(s) \) over the interval \([a, b]\) is given by \(\frac{H(b)-H(a)}{b - a}\). Here, \( a = 80 \) and \( b=100 \).

Step2: Calculate \( H(80) \)

Substitute \( s = 80 \) into \( H(s)=0.003s^{2}+0.07s - 0.027 \):
\[

$$\begin{align*} H(80)&=0.003\times(80)^{2}+0.07\times80- 0.027\\ &=0.003\times6400 + 5.6-0.027\\ &=19.2+5.6 - 0.027\\ &=24.773 \end{align*}$$

\]

Step3: Calculate \( H(100) \)

Substitute \( s = 100 \) into \( H(s)=0.003s^{2}+0.07s - 0.027 \):
\[

$$\begin{align*} H(100)&=0.003\times(100)^{2}+0.07\times100-0.027\\ &=0.003\times10000+7 - 0.027\\ &=30 + 7-0.027\\ &=36.973 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{H(100)-H(80)}{100 - 80}\):
\[

$$\begin{align*} \frac{H(100)-H(80)}{100 - 80}&=\frac{36.973 - 24.773}{20}\\ &=\frac{12.2}{20}\\ & = 0.61 \end{align*}$$

\]

Answer:

0.61