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) → \(3456\, cm^3\)
Let a square of side x cm be cut from each corner. The base of the box is a square of side \(36 - 2x\) and its height is x, where \(0 \lt x \lt 18\).
\(V = x(36 - 2x)^2\)
Differentiating by the product rule:
\( \displaystyle \frac{dV}{dx} = (36 - 2x)^2 - 4x(36 - 2x) \)
\( \displaystyle \frac{dV}{dx} = (36 - 2x)(36 - 6x) \)
In \(0 \lt x \lt 18\), \(\frac{dV}{dx} = 0\) only at x = 6. The second derivative test:
\( \displaystyle \frac{d^2V}{dx^2} = -2(36 - 6x) - 6(36 - 2x) \)
\( \displaystyle \left.\frac{d^2V}{dx^2}\right|_{x = 6} = 0 - 144 = -144 \lt 0 \)
So V is greatest at x = 6:
\(V = 6(36 - 12)^2 = 6 \times 576\)
\(V = 3456\)
The maximum volume is 3456 cm3.
Tip: For an open box made from a square sheet of side a, the volume is greatest when the cut squares have side \(\frac{a}{6}\); here \(\frac{36}{6} = 6\).
NCERT (2026-27 edition):
Application of Derivatives
