how do the average rates of change for the pair of functions compare over the given interval?\nf(x)=0.8x^2\ng…

Explanation:

Step1: Recall average rate - of - change formula

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

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

For $f(x)=0.8x^{2}$, $a = 5$, $b = 9$. $f(9)=0.8\times9^{2}=0.8\times81 = 64.8$. $f(5)=0.8\times5^{2}=0.8\times25 = 20$. The average rate of change of $f(x)$ is $\frac{f(9)-f(5)}{9 - 5}=\frac{64.8-20}{4}=\frac{44.8}{4}=11.2$.

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

For $g(x)=1.6x^{2}$, $a = 5$, $b = 9$. $g(9)=1.6\times9^{2}=1.6\times81 = 129.6$. $g(5)=1.6\times5^{2}=1.6\times25 = 40$. The average rate of change of $g(x)$ is $\frac{g(9)-g(5)}{9 - 5}=\frac{129.6 - 40}{4}=\frac{89.6}{4}=22.4$.

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 $\frac{\text{Average rate of change of }g(x)}{\text{Average rate of change of }f(x)}=\frac{22.4}{11.2}=2$.

Answer:

The average rate of change of $f(x)$ over $5\leq x\leq9$ is $11.2$. The average rate of change of $g(x)$ over $5\leq x\leq9$ is $22.4$. The average rate of change of $g(x)$ is $2$ times that of $f(x)$.