The height (in feet) of a soccer ball that was kicked is modeled by the function f(x)=−x2+9x , where x is time (in seconds) that the ball was in the air. Estimate the average rate of change over the interval [0.7,4.3] .

Answer 1

To find the average rate of change of the function \( f(x) = -x^2 + 9x \) over the interval \([0.7, 4.3]\), we will 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 \( b = 4.3 \).

1. First, we need to compute \( f(0.7) \) and \( f(4.3) \).

Calculate \( f(0.7) \): \[ f(0.7) = - (0.7)^2 + 9(0.7) = -0.49 + 6.3 = 5.81 \]

Calculate \( f(4.3) \): \[ f(4.3) = - (4.3)^2 + 9(4.3) = -18.49 + 38.7 = 20.21 \]

2. Now we can substitute these values into the average rate of change formula:

\[ \text{Average Rate of Change} = \frac{f(4.3) - f(0.7)}{4.3 - 0.7} = \frac{20.21 - 5.81}{4.3 - 0.7} \]

3. Calculate the numerator and the denominator:

Numerator: \[ 20.21 - 5.81 = 14.4 \]

Denominator: \[ 4.3 - 0.7 = 3.6 \]

4. Now plug these values into the average rate of change formula:

\[ \text{Average Rate of Change} = \frac{14.4}{3.6} = 4 \]

Thus, the average rate of change of the height of the soccer ball over the interval \([0.7, 4.3]\) is \(\boxed{4}\) feet per second.