A square piece of tin of side $18{\rm{ cm}}$ is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Let $x{\rm{cm}}$ be the length of each side of the square which is to be cut off from each corner of the square tin sheet of side $18{\rm{ cm}}$.

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

Differentiating (i) w.r.t. $x$, we get

$\cfrac{{dV}}{{dx}} = 4(3{x^2} - 36x + 81) = 12({x^2} - 12x + 27)$
For maximum/minimum volume,
$\cfrac{{dV}}{{dx}} = 0 \Rightarrow 12({x^2} - 12x + 27) = 0$

$\Rightarrow 12(x - 3)(x - 9) = 0 \Rightarrow x = 3,9$ but $0 < x < 9 \Rightarrow x = 3$

$\cfrac{{{d^2}V}}{{d{x^2}}} = 12(2x - 12) = 24(x - 6)$
and ${\left( {\cfrac{{{d^2}V}}{{d{x^2}}}} \right)_{x = 3}} = 24(3 - 6) = - 72 < 0$

$\Rightarrow V$ has a maximum value at $x = 3$.

Hence we can say that the volume of the box is maximum when the side of the square to be cut off is $3{\rm{ cm}}$.