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

Question

. 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 function \( h(t) = -16t^2 + 160t \) over the interval \([4, 5]\), we use the formula for the 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 \( h(4) \): \[ h(4) = -16(4^2) + 160(4) \] \[ = -16(16) + 640 \] \[ = -256 + 640 \] \[ = 384 \]

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

Now, substitute \( h(4) \) and \( h(5) \) into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{h(5) - h(4)}{5 - 4} = \frac{400 - 384}{1} = \frac{16}{1} = 16 \]

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