Question

Tyreke is an engineer and needs to design a rocket for an experiment. He has calculated that the height of the rocket, in feet, with respect to time, in seconds, can be modeled by the function h(t)=−16t2+160t
. Find the average rate of change of the rocket over the interval [4,5]
. (1 point)
The average rate of change is
feet per second.

Answer 1

To find the average rate of change of the rocket's height over the interval \([4, 5]\), we can use the formula for the average rate of change, which is given by:

\[
\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}
\]

where \(a\) and \(b\) are the endpoints of the interval. In this case, \(a = 4\) and \(b = 5\).

First, we need to calculate \(h(4)\) and \(h(5)\) using the function \(h(t) = -16t^2 + 160t\).

1. Calculate \(h(4)\):

\[
h(4) = -16(4)^2 + 160(4)
\]
\[
h(4) = -16(16) + 640
\]
\[
h(4) = -256 + 640
\]
\[
h(4) = 384
\]

2. Calculate \(h(5)\):

\[
h(5) = -16(5)^2 + 160(5)
\]
\[
h(5) = -16(25) + 800
\]
\[
h(5) = -400 + 800
\]
\[
h(5) = 400
\]

Now, we can calculate the average rate of change over the interval \([4, 5]\):

\[
\text{Average Rate of Change} = \frac{h(5) - h(4)}{5 - 4} = \frac{400 - 384}{5 - 4} = \frac{16}{1} = 16 \text{ feet per second.}
\]

Thus, the average rate of change of the rocket over the interval \([4, 5]\) is **16 feet per second**.