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The horsepower, [tex]H(s)[/tex], required for a racecar to overcome wind resistance is given by the function [tex]H(s)=0.003 s^2+0.07 s-0.027[/tex], where [tex]s[/tex] 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?

A. 1.64
B. 20.0
C. 12.2
D. 0.61



Answer :

GinnyAnswer
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].

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