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, \(a = 1\), \(b=3\), and \(h(t)=-16t^{2}+91t + 30\).

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

• For \(t = 1\):

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

• For \(t = 3\):

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

Step3: Apply the average rate of change formula

\(\frac{h(3)-h(1)}{3 - 1}=\frac{159 - 105}{2}=\frac{54}{2}=27\)

Answer:

27 feet per second