Chapter 10.5, Problem 27P

Stacey is designing a soup can. It must hold 32 cubic inches of soup and the radius must be approximately half the height. The function $r=\sqrt{\frac{32}{\pi h}}$ shows the radius of the can as a function of its height. What are the domain and range of the function? Graph the function. What height should Stacey design die can?

To determine

To find: the domain and range of function.

Answer to Problem 27P

The domain and range of function is $h>0\text{ and }\left(0,\infty \right)$ .

Explanation of Solution

Given information:

The function is $r=\sqrt{\frac{32}{\pi h}}$ .

Calculation: to find the domain of the function, the function is defined for all the $h$ which,

  $\begin{array}{l}\sqrt{\frac{32}{\pi h}}\ge 0\\ \frac{32}{\pi h}\ge 0\text{ [square]}\\ h>0\end{array}$

Thus, the domain of the function is $h>0$ .

And the range of the function is $\left(0,\infty \right)$ .

The graph of the function can be obtained as:

  

To find the height for design substitute $r=\frac{h}{2}$ and then simplify,

  $\begin{array}{l}r=\sqrt{\frac{32}{\pi h}}\\ \frac{h}{2}=\sqrt{\frac{32}{\pi h}}\text{ [substitute]}\\ \frac{h^2}{4}=\frac{32}{\pi h}\text{ [square]}\\ h^3=\frac{128}{\pi }\text{ [cross multiplication]}\\ h=1.67\text{ [simplify]}\end{array}$