An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

#### Solution

Let each side of the square cut off from each corner be x cm.

Then the base of the box will be of side 18 - 2x cm and the height of the box will be x cm

Then volume of box V = ( 18 - 2x )( 18 - 2x )

V = ( 18 - 2x )^{2}x

V = 4x3 + 324x - 72x2 ..(i)

Differentiating w.r t to x , we get

`(dV)/dx = 12x^2 + 324 - 144x`

`(dV)/dx = 12(x^2 - 12x + 27)` ...(ii)

For maximum volume `(dV)/dx = 0`

⇒ 12( x^{2} - 12x + 27 ) = 0

⇒ x^{2} - 9x -3x + 27 = 0

⇒ ( x - 9 )( x - 3 ) = 0

⇒ x = 9, 3

Again differentiating, we get

`(d^2V)/dx^2 = 2x - 12` ....(iii)

At x = 9.

`(d^2V)/dx^2 = +ve`

$∴$ V is minimum at x = 9 at x = 3.

`(d^2V)/dx^2 = - ve`

$∴$ V is maximum at $∴$= 12 x 12 x 3 = 432 cm^{3}