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.
Remaining Attempts : 3

Answer 1

To find the average rate of change of the height of the rocket 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 \(b = 5\).

Now, we will calculate \(h(4)\) and \(h(5)\) using the given function \(h(t) = -16t^2 + 160t\).

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

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

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

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

Now that we have \(h(4) = 384\) and \(h(5) = 400\), we can substitute these values into the average rate of change formula:

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

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