Consider the side of the square box be $\"\"$ .

The height of the box be $\"\"$ .

The surface area of the square box with open top is $\"\"$ .

But area of the box is $\"\"$ cm².

$\"\"$

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

$\"\"$

Volume of the square box is maximum when its first derivative becomes zero.

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

$\"\"$

$\"\"$ .

Equate $\"\"$ to zero.

$\"\"$

The side of the square base is $\"\"$ .

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

$\"\"$

Therefore the volume of the box has a maximum value is $\"\"$ .

The volume of the box has a maximum value is $\"\"$ .