how do the average rates of change for the pair of functions compare over the given interval? f(x)=0.9x²…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate average rate of change of $f(x)$

For $f(x)=0.9x^{2}$, $a = 4$, $b = 8$. $f(4)=0.9\times4^{2}=0.9\times16 = 14.4$ $f(8)=0.9\times8^{2}=0.9\times64 = 57.6$ The average rate of change of $f(x)$ is $\frac{f(8)-f(4)}{8 - 4}=\frac{57.6-14.4}{4}=\frac{43.2}{4}=10.8$

Step3: Calculate average rate of change of $g(x)$

For $g(x)=1.8x^{2}$, $a = 4$, $b = 8$. $g(4)=1.8\times4^{2}=1.8\times16 = 28.8$ $g(8)=1.8\times8^{2}=1.8\times64 = 115.2$ The average rate of change of $g(x)$ is $\frac{g(8)-g(4)}{8 - 4}=\frac{115.2-28.8}{4}=\frac{86.4}{4}=21.6$

Step4: Find the ratio of the average rates of change

To find how many times the average rate of change of $g(x)$ is that of $f(x)$, we calculate $\frac{21.6}{10.8}=2$

Answer:

The average rate of change of $f(x)$ over $4\leq x\leq8$ is $10.8$. The average rate of change of $g(x)$ over $4\leq x\leq8$ is $21.6$. The average rate of change of $g(x)$ is $2$ times that of $f(x)$.