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