You want to make cylindrical containers to hold 1 liter (1000\(\text {cm}^3\)) using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side \(2r\), so that \(2(2r)^2=8r^2\) of material is needed (rather than \(2\pi r^2\), which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material.
The total area of the material spent consists of the
area of the lateral side of the cylinder plus the area of two squares of side
\(2r\).
![[Picture]](singVarCalcOptimization9.online-766d31d3d528a4128d1897df7396881b.svg)
\(S= 2\pi r h +8 r^2\)
Use the fact that the volume of the cylinder, \(V=r^2\cdot \pi \cdot h\) and that \(V=1000\). Express \(h\) in terms of \(r\). Now
you can express \(S\) as a function of \(r\).
Our objective function \(S\) is given by the
expression
\(S(r)=\frac {\answer {2000}}{r}+8r^{\answer {2}}\).
\[ \text {radius}=\answer {5}\text {cm}\qquad \text {height}=\answer {40/\pi }\text {cm} \]