A cone-shaped drinking cup is drawn from the side view.

$\"\"$

Observe the figure.

From the pythagorean theorem,

$\"\"$

The volume of the cone is $\"\"$ .

$\"\"$

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

The volume of the cone is maximum when its first derivative becomes zero.

Equate $\"\"$ to zero.

$\"\"$

The height of the cone is $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore the maximum capacity of the cup is $\"\"$ .

The maximum capacity of the cup is $\"\"$ .