Consider radius of the cylinder be $\"\"$ and height of the cylinder be $\"\"$ .

The volume of the cylinder with out top is $\"\"$ .

Hence $\"\"$ .

Surface area = Area of bottom + Area of side.

Surface area = Area of bottom + (perimeter of the side * height).

The surface area of the cylinder with out top is $\"\"$ .

$\"\"$

$\"\"$ .

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

$\"\"$

$\"\"$ .

The surface area of the cylinder is maximum when its first derivative becomes zero.

Equate $\"\"$ to zero.

$\"\"$

Therefore the radius of the cylinder is $\"\"$ .

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

$\"\"$

Therefore the height of the cylinder is $\"\"$ .

The radius of the cylinder is $\"\"$ .

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