Find the maximum capacity (volume) of such a cup

Question

A cone-shaped cup is made from a circular piece of paper of radius 4 by cutting out a sector and joining the edges AC and BC.

Find the maximum capacity (volume) of such a cup

 

Answer 1

Let r be the radius of the base of the cone and h = the height of the cone.

When folded together, the side of the cone is the radius of the original circle = 4.

A right triangle is formed with the hypotenuse being 4. Thus h²+r² = 16. r² = 16-h²

The volume of the cone, V = (1/3)πr²h = (1/3)π(16-h²)h

V '(h) = (π/3)(16-3h²)

Setting the derivative to zero and solving for h²,

16-3h² = 0 --> h² = 16/3 --> r² = 16-16/3 = 32/3

Maximum volume = (1/3)π(32/3)4√3/3 = (128√3)π/27 = 25.79