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 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?

a. 1.64
b. 20.0
c. 12.2
d. 0.61

Answer 1

To find the average rate of change of horsepower per unit speed as the racecar increases in speed from 80 mph to 100 mph, we use the formula for the average rate of change, which is given by:

\[
\text{Average Rate of Change} = \frac{H(s_2) - H(s_1)}{s_2 - s_1}
\]

where \(s_1 = 80\) mph and \(s_2 = 100\) mph.

First, we need to calculate \(H(80)\) and \(H(100)\) using the given function \(H(s) = 0.003s^2 + 0.07s - 0.027\).

**Calculating \(H(80)\):**

\[
H(80) = 0.003(80)^2 + 0.07(80) - 0.027
\]
\[
H(80) = 0.003(6400) + 5.6 - 0.027
\]
\[
H(80) = 19.2 + 5.6 - 0.027
\]
\[
H(80) = 24.772
\]

**Calculating \(H(100)\):**

\[
H(100) = 0.003(100)^2 + 0.07(100) - 0.027
\]
\[
H(100) = 0.003(10000) + 7 - 0.027
\]
\[
H(100) = 30 + 7 - 0.027
\]
\[
H(100) = 36.973
\]

Now we can find the average rate of change:

\[
\text{Average Rate of Change} = \frac{H(100) - H(80)}{100 - 80}
\]
\[
\text{Average Rate of Change} = \frac{36.973 - 24.772}{20}
\]
\[
= \frac{12.201}{20} = 0.61005
\]

Thus, the average rate of change is approximately \(0.61\) (rounded to two decimal places).

So the answer is:

**d. 0.61**