A circular cylinder needs to hold a volume of 4.6 square meters. Find the dimensions that minimize the amount of material needed, that is, minimize the surface area.

The objective is to minimize the surface area. Drawing a diagram and decomposing the surface into two circles and a rectangle gives: The area of each circle is $LaTeX: \displaystyle A_c = \pi r^2$ and the area of the rectangle is $LaTeX: \displaystyle A_r = 2\pi r h$ . The function to be minimized is $LaTeX: \displaystyle f = 2\pi r^2+2\pi r h$ . The constraint on the volume can be used to eliminate $LaTeX: \displaystyle h$ from the function $LaTeX: \displaystyle f$ . Solving $LaTeX: \displaystyle 4.6 = \pi r^2 h$ for $LaTeX: \displaystyle h$ gives $LaTeX: \displaystyle h = \frac{4.6}{\pi r^{2}}$ Substituting $LaTeX: \displaystyle h$ into $LaTeX: \displaystyle f$ gives $LaTeX: \displaystyle f = 2\pi r^2+2\pi r \left( \frac{4.6}{\pi r^2}\right) = 2\pi r^2 + \frac{9.2}{r}$ . Taking the derivative gives $LaTeX: \displaystyle f'(r) = 4 \pi r - \frac{9.2}{r^{2}}$ . Clearing the fractions gives $LaTeX: \displaystyle 9.2=4 \pi r^{3} \iff r^3=2.3 \iff r = \frac{\sqrt[3]{2.30}}{\sqrt[3]{\pi}}$ Substituing back into the equation for $LaTeX: \displaystyle h$ gives $LaTeX: \displaystyle h = \sqrt[3]{\frac{18.4}{\pi}}$