Largest dimension of cone

Question

A right circular cone is inscribed in a sphere of radius 2 units. What will be the dimensions of the cone with largest volume? JUSTIFY your answer

Answer 1

Let z: distance from the circular top of the cone to the xy-plane

z² + r² = 2² so r = √(4 - z²)

V(z) = 1/3·π·(4 - z²)(2 + z) = π/3 · [ - z³ - 2z² + 4z + 8 ]

V^'(z) = π/3 · [ - 3z² - 4z + 4 ] = 0

z = [4 ± √(64)] / - 6 = 2/3 and V^' goes from + to - there so V(2/3) is a maximum.

r = √(32/9) = 4√2 / 3 and h = 8/3