Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle `alpha,` is `(4)/(27) pi h^(3) tan^(2) alpha`.

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Prove that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is (2R)/sqrt(3) . Also find the maximum volume .

A solid consisting of a right circular cone standing on a hemisphere , is placed upright in a right circular cylinder full of water and touching the bottom. Find the volume of water left in the cylinder , given that the radius of the cylinder is 3 cm and its height is 6 cm The radius of the hemisphere is 2 cm and the height of the cone is 4 cm . (Take pi =(22)/(7) ).

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be a and p. If the height of the lighthouse is h m etres and the line joining the ships passes through the foot of the lighthouse, show that the distance betw een the ships is (h(tan alpha + tan beta))/(tan alpha tan beta)

From the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base of the original cone. Find the total surface area of the remaining solid. [use pi=3.14,sqrt(5)=2.236 ]

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan^(-1)(0.5) . Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m.