To find the average rate of change of the function \( h(t) = -16t^2 + 160t \) over the interval \([4, 5]\), we will 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 \( h(5) \).
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Calculate \( h(4) \): \[ h(4) = -16(4)^2 + 160(4) \] \[ = -16(16) + 640 \] \[ = -256 + 640 = 384 \]
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Calculate \( h(5) \): \[ h(5) = -16(5)^2 + 160(5) \] \[ = -16(25) + 800 \] \[ = -400 + 800 = 400 \]
Now, we can find the average rate of change using these values:
\[ \text{Average Rate of Change} = \frac{h(5) - h(4)}{5 - 4} \] \[ = \frac{400 - 384}{5 - 4} = \frac{16}{1} = 16 \]
Therefore, the average rate of change of \( h(t) \) over the interval \([4, 5]\) is \( 16 \).