91Ó°ÊÓ

A cylinder is cut from a solid sphere of radius \(5 \mathrm{~cm}\). If the height of the cylinder is \(2 h\), show that the volume of the cylinder is \(2 \pi h\left(25-h^{2}\right)\), assuming that the curved edges of the cylinder reach the surface of the sphere. Find the maximum volume of such a cylinder.

If a piece of string of fixed length is made to enclose a rectangle, show that the enclosed area is greatest when the rectangle is a square.

Find the maximum displacement of a particle from a point \(\mathrm{O}\), if its displacement \(s\) metres from \(\mathrm{O}\) after time \(t\) seconds is given by $$ s=2+3 t-t^{2} $$