Find the angle of intersection of the curves 2y² = x³ and y² = 32x.

Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).

Sand is pouring from a pipe at the rate of 12 cm³/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing, when the height is 4 cm?

A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan⁻¹(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.