Chapter 5.3, Problem 5GP

Solution

To graph: The dimensions of the can that is the height and the radius of the can.

The dimensions of the can will be height $h=9cm$ and radius $r=4.424cm$.

Write the equation that gives the height $h$ of the can in terms of its radius $r$ . Use the formula for the volume of a cylinder and the fact that the soup can’s volume is $544$ cubic centimeters.

  $\begin{array}{l}V=\pi r^{2}h\\ 544=\pi r^{2}h\\ h=\frac{544}{\pi r^2}\end{array}$

Write the function that gives the surface area of the can in terms of only its radius

  $\begin{array}{l}S=2\pi r^2+2\pi rh\\ =2\pi r^2+2\pi r\frac{544}{\pi r^2}\\ =2\pi r^2+\frac{1088}{r}\end{array}$

Graph the function for the surface area using a graphing calculator. Then use the minimum feature to find the minimum value of surface area.

You get a minimum value of about $368.90$ , which occurs when $r=4.424$ and

  $\begin{array}{l}h=\frac{544}{\pi r^2}\\ h=\frac{544}{\pi {(4.424)}^2}\\ h=\frac{544}{61.474948416}\\ h=8.849\\ h\approx 9\end{array}$