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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. 12.2
C. 0.61
D. 20.0



Answer :

GinnyAnswer
Given the function [tex]\( H(s) = 0.003s^2 + 0.07s - 0.027 \)[/tex], we need to find the average rate of change in horsepower per unit speed as the speed increases from 80 mph to 100 mph.

### Step-by-Step Solution

1. Evaluate [tex]\( H(s) \)[/tex] at [tex]\( s = 80 \)[/tex] mph:
[tex]\[ H(80) = 0.003 \times 80^2 + 0.07 \times 80 - 0.027 \][/tex]

2. Calculate [tex]\( 80^2 \)[/tex]:
[tex]\[ 80^2 = 6400 \][/tex]

3. Substitute [tex]\( 6400 \)[/tex] and other values into the function and simplify:
[tex]\[ H(80) = 0.003 \times 6400 + 0.07 \times 80 - 0.027 \][/tex]
[tex]\[ H(80) = 19.2 + 5.6 - 0.027 \][/tex]
[tex]\[ H(80) = 24.773 \][/tex]

4. Evaluate [tex]\( H(s) \)[/tex] at [tex]\( s = 100 \)[/tex] mph:
[tex]\[ H(100) = 0.003 \times 100^2 + 0.07 \times 100 - 0.027 \][/tex]

5. Calculate [tex]\( 100^2 \)[/tex]:
[tex]\[ 100^2 = 10000 \][/tex]

6. Substitute [tex]\( 10000 \)[/tex] and other values into the function and simplify:
[tex]\[ H(100) = 0.003 \times 10000 + 0.07 \times 100 - 0.027 \][/tex]
[tex]\[ H(100) = 30 + 7 - 0.027 \][/tex]
[tex]\[ H(100) = 36.973 \][/tex]

7. Calculate the average rate of change in horsepower per unit speed:
[tex]\[ \text{Average Rate of Change} = \frac{H(100) - H(80)}{100 - 80} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{36.973 - 24.773}{20} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{12.2}{20} \][/tex]
[tex]\[ \text{Average Rate of Change} = 0.61 \][/tex]

Thus, the average rate of change in horsepower per unit speed when the racecar increases its speed from 80 mph to 100 mph is [tex]\( 0.61 \)[/tex] horsepower per mph.

So the correct answer is:
[tex]\[ \boxed{0.61} \][/tex]

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