The given figure represents a cylinder Inscribed in a sphere. 1. Find an expression for the volume V of the cylinder.

Question

The given figure represents a cylinder Inscribed in a sphere.

1. Find an expression for the volume V of the cylinder.
2. Find the height of the cylinder when its volume V is maximum.
3. Find the volume and radius of the largest cylinder.

Answer 1

 1. From the right triangle ∆OAB,

y² = R² – x² ⇒ y = \(\sqrt{R^2 - x^2}\)

Which is the radius of the cylinder

Also height = 2x

∴ Volume = V= π y² × 2x = 2π(R² – x²)x = 2π(R²x – x³).

2. Now,

For maximum or minimum,

3. Base radius =