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] .

Answer 1

To find the average rate of change of the rocket's height \( h(t) \) over the interval \([4, 5]\), we will use the formula for the average rate of change:

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

where \( a = 4 \), \( b = 5 \), and \( h(t) = -16t^2 + 160t \).

First, we need to evaluate \( h(4) \) and \( h(5) \).

**Calculate \( h(4) \):**

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

**Calculate \( h(5) \):**

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

Now we can use these values to find the average rate of change over the interval \([4, 5]\):

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

Thus, the average rate of change of the rocket's height over the interval \([4, 5]\) is \( \boxed{16} \) feet per second.