Find the Volume of a Cylinder Inscribed in a Sphere

Problem:

A right circular cylinder of radius 'r' is inscribed in a sphere of radius '4r'. Find a formula for 'V', the volume of the cylinder, in terms of 'r'.

Solution:

The volume of a cylinder is given by the formula V = πr^2h, where 'r' is the radius of the cylinder and 'h' is its height.

In this case, the cylinder is inscribed in a sphere of radius '4r'. Let's consider the cross-section of the cylinder and sphere:

• The diameter of the cylinder is equal to the diameter of the sphere, which is '8r'.
• The height of the cylinder is the distance between the top and bottom of the sphere, which is '2 * 4r = 8r'.

Since the diameter of the cylinder is '8r', the radius of the cylinder is '4r'. Now we can calculate the volume of the cylinder:

V = πr^2h = π * (4r)^2 * 8r = 128πr^3

Therefore, the formula for the volume of the cylinder in terms of 'r' is V = 128πr^3