Chapter 3.7, Problem 28E

Numerical, Graphical, and Analytic Analysis A right circular cylinder is designed to hold 22 cubic inches of a soft drink (approximately 12 fluid ounces).

(a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.)

Radius, r	Height	Surface Area, S
0.2	$\frac{22}{\pi {(0.2)}^2}$	$2\pi (0.2)\left[0.2+\frac{22}{\pi {(0.2)}^2}\right]\approx 220.3$
0.2	$\frac{22}{\pi {(0.4)}^2}$	$2\pi (0.4)\left[0.4+\frac{22}{\pi {(0.4)}^2}\right]\approx 111.0$

(b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the minimum surface area.

(c) Write the surface area S as a function of r.

(d) Use calculus to find the critical number of the function in part (c). Then find the minimum surface area and the dimensions that yield the minimum surface area.

(e) Use a graphing utility to graph the function in part (c) and verify the minimum surface area from the graph.