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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 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?



Answer :

adeletedaccount

To find the average rate of change in horsepower per unit speed, we need to calculate the change in horsepower (ΔH) divided by the change in speed (Δs) over the given speed interval.

First, let's find the horsepower at 80 mph and 100 mph:

- At 80 mph:

 H(80) = 0.003(80)^2 + 0.07(80) - 0.027

        = 0.003(6400) + 5.6 - 0.027

        = 19.2 + 5.6 - 0.027

        = 24.773 horsepower

- At 100 mph:

 H(100) = 0.003(100)^2 + 0.07(100) - 0.027

         = 0.003(10000) + 7 - 0.027

         = 30 + 7 - 0.027

         = 37.973 horsepower

Now, we calculate the average rate of change:

ΔH = H(100) - H(80) = 37.973 - 24.773 = 13.2 horsepower

Δs = 100 mph - 80 mph = 20 mph

Average rate of change = ΔH / Δs = 13.2 horsepower / 20 mph ≈ 0.66 horsepower per mph

So, the average rate of change in horsepower per unit speed as the racecar increases in speed from 80 mph to 100 mph is approximately 0.66 horsepower per mph.

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