Question

A conical tent of the given capacity (volume) has to be constructed. Find the ratio of the
height to the radius of the base so as to minimise the canvas required for the tent.
Solve using Maxima/Minima of second order derivative. Pls help.

Answer 1

So you want to minimize the amount of material required to create a conical tent with a fixed volume. The amount of material required is just the surface area function.

\[V = \frac {\pi}{3} r^2 h, A = \pi r^2 + \pi r \sqrt {h^2 + r^2}\]You can solve for either r or h in the volume equation and substitute that into the surface area function. Then, you can take the derivative and find the max radius/height that will produce that constant volume V.