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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.