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
Answer & explanation
Correct answer: option 4
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 $= 6(36 - 12)^2$
$= 6(24)^2$
$= 3456 \text{ cm}^3$