A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also find this maximum volume.

Side to be cut off = 3 cm, Maximum Volume = 432 cm$^3$

The correct answer is Option (2) → Side to be cut off = 3 cm, Maximum Volume = 432 cm$^3$

Let $x\, cm\, (0 < x <9)$ be the length of each side of the square which is to be cut off from corners of the square tin sheet of side 18 cm.

Let V be the volume of the open box formed by folding up the flaps, then $V = (18-2x) (18-2x) x=4(9-x)^2 x = 4(x^3- 18x^2 +81x)$.

Differentiating it w.r.t. x, we get

$\frac{dV}{dx}=4(3x^2-36x+81) = 12 (x^2 - 12x+27)$ and

$\frac{d^2V}{dx^2}= 12 (2x-12) = 24(x-6)$.

Now $\frac{dV}{dx}=0⇒12(x^2-12x+27) = 0⇒x^2-12x+27=0$

$⇒(x-3)(x-9) = 0$ but $0 < x <9⇒ x=3$.

Also, at $x = 3, \frac{d^2V}{dx^2}=24(3-6)=-72 <0$ $⇒ V$ is maximum when $x = 3$.

Therefore, the volume of the box is maximum when the side of the square to be cut off is 3 cm.

Also maximum volume = $4(9-3)2.3\, cm^3 = 432\, cm^3$.