Minimizing the volume of a cylinder using fixed surface area

Question

Considering that I want my cylinder to have a fixed surface area of 0.3m^2, how can I minimize the volume of the cylinder. I have already tried to optimize it but I am only able to maximize the volume.

Answer 1

In the limits, the volume approaches 0, there technically isn't a minimum in the sense of 'local extrema'. Consider the following:

$$A=2\pi( r^2 + rh)=0.3$$

$$h =\frac{0.3}{2\pi r}-r$$

$$V=\pi r^2 h = \pi r^2 (\frac{0.3}{2\pi r}-r)=\frac{0.3r}{2} - \pi r^3$$

Notice as $r \rightarrow 0$, $V \rightarrow 0$, and of course $h \rightarrow \infty$.