A right circular cylinder of radius $a$ and length $l$ has volume $V$ and total surface area $A$. Use Lagrange multipliers to do the following:

(a) Show that, for a given total surface area, the maximum volume is

$$V=\frac{1}{3} \sqrt{\frac{A^{3}}{C \pi}}$$

determining the integer $C$ in the process.

(b) For a cylinder inscribed in the unit sphere, show that the value of $l / a$ which maximises the area of the cylinder is

$$D+\sqrt{E}$$

determining the integers $D$ and $E$ as you do so.

(c) Consider the rectangular parallelepiped of largest volume which fits inside a hemisphere of fixed radius. Find the ratio of the parallelepiped's volume to the volume of the hemisphere.

[You need not show that suitable extrema you find are actually maxima.]