Techniques and Applications Of Differentiation Q5

A solid right circular cylinder is to be cut out of a solid right circular cone, whose radius is \({1.5}\)m and its height is \({3}\)m.

The axis of symmetry of the cone coincides with the axis of symmetry of the cylinder which passes through its circular ends. The circumference of one end of the cylinder is in contact with the curve surface of the cone and the other end of the cylinder lies on the base of the cone.

Show that the maximum volume of the cylinder to be cut out is \({\pi }\) m\({^{3}}\).