Question

practice: average rate of change
for each scenario, determine the average rate of change over the given interval. then interpret the rate of change in context using appropriate units.

1. the path of a baseball after it has been hit is modeled by the function ( h = - 0.0032d^{2}+d + 3 ), where ( h ) is the height (in feet) of the baseball and ( d ) is the distance (in feet) the baseball is from home plate. determine the average rate of change in height on the interval ( 30leq dleq130 ) feet. interpret the rate in context.

Explanation:

Step1: Find \(h(30)\)

Substitute \(d = 30\) into \(h(d)=-0.0032d^{2}+d + 3\)

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Step2: Find \(h(130)\)

Substitute \(d = 130\) into \(h(d)=-0.0032d^{2}+d + 3\)

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Step3: Calculate the average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, \(x_1 = 30\), \(x_2=130\), \(f(d)=h(d)\)

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LATEXBLOCK2
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Answer:

The average rate of change is \(0.488\) feet per foot. This means that, on average, as the baseball travels from \(30\) feet to \(130\) feet from home - plate, its height increases at a rate of \(0.488\) feet for every one - foot increase in its distance from home plate.