Question
The horsepower, Hs), required for a racecar to overcome wind resistance is given by the
function H(s) = 0.003s? + 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?
function H(s) = 0.003s? + 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?
Answers
GPT-4o mini
To find the average rate of change in horsepower \( H(s) \) per unit speed from 80 mph to 100 mph, we will use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{H(b) - H(a)}{b - a}
\]
where \( a = 80 \) mph and \( b = 100 \) mph.
First, we need to calculate \( H(80) \) and \( H(100) \).
1. Calculate \( H(80) \):
\[
H(80) = 0.003(80)^2 + 0.07(80) - 0.027
\]
Calculating each term:
\[
(80)^2 = 6400 \implies 0.003(6400) = 19.2
\]
\[
0.07(80) = 5.6
\]
So:
\[
H(80) = 19.2 + 5.6 - 0.027 = 24.773
\]
2. Calculate \( H(100) \):
\[
H(100) = 0.003(100)^2 + 0.07(100) - 0.027
\]
Calculating each term:
\[
(100)^2 = 10000 \implies 0.003(10000) = 30
\]
\[
0.07(100) = 7
\]
So:
\[
H(100) = 30 + 7 - 0.027 = 36.973
\]
Now we can find the average rate of change:
\[
\text{Average Rate of Change} = \frac{H(100) - H(80)}{100 - 80} = \frac{36.973 - 24.773}{20}
\]
Calculating the difference in horsepower:
\[
36.973 - 24.773 = 12.2
\]
Now we divide by the difference in speed:
\[
\text{Average Rate of Change} = \frac{12.2}{20} = 0.61
\]
Thus, the average rate of change in horsepower per unit speed when the racecar increases in speed from 80 mph to 100 mph is \( \boxed{0.61} \) horsepower per mph.
\[
\text{Average Rate of Change} = \frac{H(b) - H(a)}{b - a}
\]
where \( a = 80 \) mph and \( b = 100 \) mph.
First, we need to calculate \( H(80) \) and \( H(100) \).
1. Calculate \( H(80) \):
\[
H(80) = 0.003(80)^2 + 0.07(80) - 0.027
\]
Calculating each term:
\[
(80)^2 = 6400 \implies 0.003(6400) = 19.2
\]
\[
0.07(80) = 5.6
\]
So:
\[
H(80) = 19.2 + 5.6 - 0.027 = 24.773
\]
2. Calculate \( H(100) \):
\[
H(100) = 0.003(100)^2 + 0.07(100) - 0.027
\]
Calculating each term:
\[
(100)^2 = 10000 \implies 0.003(10000) = 30
\]
\[
0.07(100) = 7
\]
So:
\[
H(100) = 30 + 7 - 0.027 = 36.973
\]
Now we can find the average rate of change:
\[
\text{Average Rate of Change} = \frac{H(100) - H(80)}{100 - 80} = \frac{36.973 - 24.773}{20}
\]
Calculating the difference in horsepower:
\[
36.973 - 24.773 = 12.2
\]
Now we divide by the difference in speed:
\[
\text{Average Rate of Change} = \frac{12.2}{20} = 0.61
\]
Thus, the average rate of change in horsepower per unit speed when the racecar increases in speed from 80 mph to 100 mph is \( \boxed{0.61} \) horsepower per mph.