Chapter 10.1, Problem 49E

a.

To determine

To graph: The function using graphing calculator. Also, find the pressure at which the flow rate exceeds 300 gallons per minute.

Explanation of Solution

Given information:

The model of the flow rate: $f=120\sqrt{p}$

The given equation for flow rate is $f=120\sqrt{p}$ .

The graph of the function is shown below:

  

From the above graph, it is seen that flow rate will exceed 300 gallons per minute at a pressure of 6.25 pounds per square inch.

b.

To determine

To write: The description about what happens to the average rate of change of the flow as the pressure increases.

Answer to Problem 49E

The average rate of changes of the flow rate decreases as the pressure increases.

Explanation of Solution

Given information:

The model of the flow rate: $f=120\sqrt{p}$

The given equation for flow rate is $f=120\sqrt{p}$ .

Here, find the average rates of change over the intervals $x=0$ to $x=2$ , $x=2$ to $x=4$ and $x=4$ to $x=6$ .

For the interval $x=0$ to $x=2$ ;

The average rate of change is,

  $\begin{array}{c}\frac{f\left(2\right)-f\left(0\right)}{2-0}=\frac{120\sqrt{2}-120\sqrt{0}}{2-0}\\ \approx 84.85\end{array}$

For the interval $x=2$ to $x=4$ ;

The average rate of change is,

  $\begin{array}{c}\frac{f\left(4\right)-f\left(2\right)}{4-2}=\frac{120\sqrt{4}-120\sqrt{2}}{4-2}\\ \approx 35.15\end{array}$

For the interval $x=4$ to $x=6$ ;

The average rate of change is,

  $\begin{array}{c}\frac{f\left(6\right)-f\left(4\right)}{6-4}=\frac{120\sqrt{6}-120\sqrt{4}}{6-4}\\ \approx 26.97\end{array}$

Hence, the average rate of changes of the flow rate decreases as the pressure increases.