To find the average rate of change of the rocket's height 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 \(b = 5\).
First, we need to calculate \(h(4)\) and \(h(5)\) using the function \(h(t) = -16t^2 + 160t\).
- Calculate \(h(4)\):
\[ h(4) = -16(4)^2 + 160(4) \] \[ h(4) = -16(16) + 640 \] \[ h(4) = -256 + 640 \] \[ h(4) = 384 \]
- Calculate \(h(5)\):
\[ h(5) = -16(5)^2 + 160(5) \] \[ h(5) = -16(25) + 800 \] \[ h(5) = -400 + 800 \] \[ h(5) = 400 \]
Now, we can calculate 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.