Question

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.
On the basis of the above information, answer the following questions :
Find a relation between \( x \) and \( y \) such that the surface area \( S \) is minimum.

Hint:

To minimize the surface area, set the derivative of \( S \) to zero and solve for \( x \), then substitute into the equation for \( y \).

Answer 1

To minimize surface area, set \( \frac{dS}{dx} = 0 \). This yields: \[ 2x - \frac{4V}{x^2} = 0 \] Solving for \( x \): \[ 2x = \frac{4V}{x^2} \quad \Rightarrow \quad 2x^3 = 4V \quad \Rightarrow \quad x^3 = 2V \] Therefore, \( x = \sqrt[3]{2V} \). Substituting this value of \( x \) into the equation for \( y \): \[ y = \frac{V}{x^2} = \frac{V}{\left(\sqrt[3]{2V}\right)^2} = \frac{V}{\sqrt[3]{(2V)^2}} = \frac{V}{\sqrt[3]{4V^2}} \] The relationship between \( x \) and \( y \) is thus: \[ y = \frac{V}{\sqrt[3]{4V^2}} \]