Compare the estimated average rates of change for the functions p(x) = sqrt(x) - 5...

1. Calculate the average rate of change for p(x): The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula $\frac{f(b) - f(a)}{b - a}$. For $p(x) = \sqrt{x} - 5$ over the interval $[0.1, 8.9]$, we have:
$$p(0.1) = \sqrt{0.1} - 5 \approx 0.316 - 5 = -4.684$$
$$p(8.9) = \sqrt{8.9} - 5 \approx 2.983 - 5 = -2.017$$
Average rate of change for $p(x) = \frac{p(8.9) - p(0.1)}{8.9 - 0.1} = \frac{-2.017 - (-4.684)}{8.8} = \frac{2.667}{8.8} \approx 0.303$$
2. Calculate the average rate of change for q(x): For $q(x) = 5\sqrt[3]{x} - 1$ over the interval $[0.1, 8.9]$, we have:
$$q(0.1) = 5\sqrt[3]{0.1} - 1 \approx 5(0.464) - 1 = 2.32 - 1 = 1.32$$
$$q(8.9) = 5\sqrt[3]{8.9} - 1 \approx 5(2.072) - 1 = 10.36 - 1 = 9.36$$
Average rate of change for $q(x) = \frac{q(8.9) - q(0.1)}{8.9 - 0.1} = \frac{9.36 - 1.32}{8.8} = \frac{8.04}{8.8} \approx 0.914$$
3. Compare the average rates of change: The average rate of change for $p(x)$ is approximately $0.303$, and the average rate of change for $q(x)$ is approximately $0.914$. Since $0.914 > 0.303$, the estimated average rate of change of $q(x)$ is greater than the estimated average rate of change of $p(x)$ over the interval $[0.1, 8.9]$.