To find the average rate of change of the rocket's height \( h(t) \) over the interval \([4, 5]\), we can 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 find \( h(4) \) and \( h(5) \).
- Calculate \( h(4) \):
\[ h(4) = -16(4)^2 + 160(4) \] \[ = -16(16) + 640 \] \[ = -256 + 640 \] \[ = 384 \text{ feet} \]
- Calculate \( h(5) \):
\[ h(5) = -16(5)^2 + 160(5) \] \[ = -16(25) + 800 \] \[ = -400 + 800 \] \[ = 400 \text{ feet} \]
Now we can find the average rate of change from \( t = 4 \) to \( t = 5 \):
\[ \text{Average Rate of Change} = \frac{h(5) - h(4)}{5 - 4} \] \[ = \frac{400 - 384}{1} \] \[ = \frac{16}{1} = 16 \text{ feet per second} \]
Therefore, the average rate of change is 16 feet per second.