Let's solve the problem step-by-step.
1. Define the function:
The given function for horsepower [tex]\( H(s) \)[/tex] in relation to speed [tex]\( s \)[/tex] is:
[tex]\[
H(s) = 0.003s^2 + 0.07s - 0.027
\][/tex]
2. Initial speed and horsepower:
When the speed [tex]\( s \)[/tex] is 80 mph:
[tex]\[
H(80) = 0.003(80)^2 + 0.07(80) - 0.027
\][/tex]
According to the provided information:
[tex]\[
H(80) = 24.773
\][/tex]
3. Final speed and horsepower:
When the speed [tex]\( s \)[/tex] is 100 mph:
[tex]\[
H(100) = 0.003(100)^2 + 0.07(100) - 0.027
\][/tex]
According to the provided information:
[tex]\[
H(100) = 36.973
\][/tex]
4. Average rate of change:
The average rate of change of horsepower with respect to speed from 80 mph to 100 mph is calculated using the following formula:
[tex]\[
\text{Average rate of change} = \frac{H(100) - H(80)}{100 - 80}
\][/tex]
Substituting the values we have:
[tex]\[
\text{Average rate of change} = \frac{36.973 - 24.773}{100 - 80}
\][/tex]
[tex]\[
\text{Average rate of change} = \frac{12.2}{20}
\][/tex]
[tex]\[
\text{Average rate of change} = 0.61
\][/tex]
Therefore, the average rate of change in horsepower per unit speed, as the racecar increases in speed from 80 mph to 100 mph, is [tex]\( \boxed{0.61} \)[/tex].
