Question

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.