CASE STUDY CHALLENGE
AOD: Optimization of a Wooden Box
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
Constraint: The box has a square base with length = breadth = \( x \) m and height = \( y \) m. The volume \( V \) remains constant.
Answer the following questions:
| (i) | Express the surface area \( S \) of the box in terms of \( x \) and its volume \( V \). | [1] |
| (ii) | Find \( \frac{dS}{dx} \). | [1] |
| (iii) | (a) Find a relation between \( x \) and \( y \) such that the surface area \( S \) is minimum.
— OR —
(b) If surface area \( S \) is constant, show that volume \( V = \frac{1}{4}(Sx – 2x^3) \) is maximum for \( x = \sqrt{\frac{S}{6}} \). |
[2] |