To find the average rate of change of the function \( f(x) = -x^2 + 9x \) over the interval \([0.7, 4.3]\), we use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 0.7 \) and \( b = 4.3 \).
First, we need to calculate \( f(0.7) \) and \( f(4.3) \).
1. Calculate \( f(0.7) \):
\[ f(0.7) = - (0.7)^2 + 9(0.7) = -0.49 + 6.3 = 5.81 \]
2. Calculate \( f(4.3) \):
\[ f(4.3) = - (4.3)^2 + 9(4.3) = -18.49 + 38.7 = 20.21 \]
3. Now, apply the values to the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(4.3) - f(0.7)}{4.3 - 0.7} = \frac{20.21 - 5.81}{4.3 - 0.7} \]
Calculate the difference in height and the difference in time:
\[ \text{Average Rate of Change} = \frac{20.21 - 5.81}{4.3 - 0.7} = \frac{14.4}{3.6} = 4 \]
Conclusion:
The average rate of change of the height of the soccer ball over the interval \([0.7, 4.3]\) is \( 4 \) feet per second.
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