A square board of side 36 cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

$3456\, cm^3$

The correct answer is Option (4) → $3456\, cm^3$

Side of square board = $36 \text{ cm}$

Cut square of side = $x$

Volume $V(x) = x(36 - 2x)^2$

$\frac{dV}{dx} = (36 - 2x)^2 - 4x(36 - 2x)$

$= (36 - 2x)(36 - 6x)$

Critical points: $x = 6 , 18$

$x = 18 \Rightarrow V = 0$ (not maximum)

$x = 6 \Rightarrow$ volume maximum

Maximum volume $= 6(36 - 12)^2$

$= 6(24)^2$

$= 3456 \text{ cm}^3$