Question

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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 height function \( h(t) = -16t^2 + 160t \) over the interval \([4, 5]\), we can use the formula for average rate of change:

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

where \( a = 4 \) and \( b = 5 \).

First, we need to calculate \( h(4) \) and \( h(5) \):

1. Calculate \( h(4) \):
\[
h(4) = -16(4^2) + 160(4)
\]
\[
= -16(16) + 640
\]
\[
= -256 + 640
\]
\[
= 384 \text{ feet}
\]

2. Calculate \( h(5) \):
\[
h(5) = -16(5^2) + 160(5)
\]
\[
= -16(25) + 800
\]
\[
= -400 + 800
\]
\[
= 400 \text{ feet}
\]

Now we can use these values to find 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**.