A regular square based pyramid is inscribed in a sphere of given radius `R` so that all vertices of the pyramid belong to the sphere. Find the greatest value of the volume of the pyramid.

A regular square pyramid is 3m. Height and the perimeter of its base is 16 m. Find the volume of the payment.

The lateral edge of a regular rectangular pyramid is ac mlongdot The lateral edge makes an angle alpha with the plane of the base. Find the value of alpha for which the volume of the pyramid is greatest.

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is (8)/(27) of the volume of the sphere.

The volume of a sphere is given by V= 4/3 pi R^3 where R is the radius of the sphere (a). Find the rate of change of volume with respect to R. (b). Find the change in volume of the sphere as the radius is increased from 20.0 cm to 20.1 cm. Assume that the rate does not appreciably change between R=20.0 cm to R=20.1 cm.

The volume of a cylinder is given by the formula V=pir^(2)h . Find the greatest and least values of V if r + h = 6.

The volume of cube is 1728 cubic cm. Find the volume of square pyramid of the same height.

The density inside a solid sphere of radius a is given by rho=rho_0/r , where rho_0 is the density ast the surface and r denotes the distance from the centre. Find the graittional field due to this sphere at a distance 2a from its centre.

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 2/3 of the diameter of the sphere.

A particle of mass m is kept on a fixed, smooth sphere of radius R at a position, where the radius through the particle makes an angle of 30 ∘ with the vertical. The particle is released from this position. (a) What is the force exerted by the sphere on the particle just after the release? (b) Find the distance traveled by the particle before it leaves contact with the sphere.