Question

a rocket is launched straight up from the top of a 30 - foot tall building with an initial speed of 91 feet per second. the height h(t) of the rocket can be modeled by the quadratic function h(t)=-16t² + 91t + 30, where height is measured in feet and time t is measured in seconds. which statement best describes the average rate of change in the height of the rocket from 1 to 3 seconds? 18 feet per second 27 feet per second 54 feet per second 91 feet per second

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(f(t)=h(t)=-16t^{2}+91t + 30\), \(a = 1\), and \(b = 3\).

Step2: Calculate \(h(1)\) and \(h(3)\)

• For \(t = 1\):

\(h(1)=-16\times(1)^{2}+91\times(1)+30=-16 + 91+30=105\)

• For \(t = 3\):

\(h(3)=-16\times(3)^{2}+91\times(3)+30=-16\times9 + 273+30=-144+273 + 30=159\)

Step3: Calculate the average rate of change

Using the formula \(\frac{h(3)-h(1)}{3 - 1}\), substitute \(h(1)=105\) and \(h(3)=159\)
\(\frac{159 - 105}{3 - 1}=\frac{54}{2}=27\)

Answer:

27 feet per second