Maxima and Minima

Prove that volume of largest cone, which can be inscribed in a sphere, is $\frac{8}{27}$ part of sphere.

A window is in the form of rectangle surmounted by a semi-circular opening. The perimeter of window is $30\mathrm{m}$. Find the dimensions of window so that it can admit maximum light through the whole opening.

If the height of right circular cylinder of maximum volume that can be inscribed in a sphere of radius $\sqrt{3}$ is $k\mathrm{\pi }$ cubic units, then the value of $k$ is

An open box is to be made of a square sheet of tin with side $20 \mathrm{cm}$ by cutting off small squares from each corner and folding the flaps Find the side of small square which is to be cut off so that volume of box is maximum.