A cylindrical can (with lid) is required to have a volume of 8000 cm3
. Using calculus,
determine the dimensions that minimize the surface area (and hence cost) of the can. Is
the can more costly to construct than a closed cubic container of the same volume and made
of the same material? Justify your answer.
[Math] Determining the dimensions of a cylindrical can
calculus
Best Answer
For a cylinder, $V=\pi r^2h = 8000$
Surface Area $SA = 2 \pi r^2 + 2 \pi r h$
Using the volume formula, solve for h: $$h=\frac{8000}{\pi r^2}$$
Substitute h into the surface area formula, getting $$SA=2\pi r^2 + \frac{16000\pi r}{\pi r^2}$$
Simplify to get $$SA=2\pi r^2 + \frac{16000}{r}$$
Take the derivative and set it equal to zero to find extrema:
$$SA' = 4\pi r - \frac{16000}{r^2} = 0$$
Solve for r to find the radius of the cylinder with the least SA (approximately 10.839). Use this to find the SA.
I leave the rest as an exercise for you.