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 :
Taking length = breadth = \( x \) m and height = \( y \) m, express the surface area \( S \) of the box in terms of \( x \) and its volume \( V \), which is constant.

Hint:

To express the surface area in terms of \( x \), substitute the volume equation into the surface area equation.

Answer 1

Let the length and breadth of the box be \( x \) m and the height be \( y \) m. The surface area \( S \) consists of the area of the base and the area of the four sides: \[ S = x^2 + 4xy \] Now, the volume \( V \) of the cuboid is given by: \[ V = x^2 y \] Since \( V \) is constant, we can express \( y \) in terms of \( x \) and \( V \): \[ y = \frac{V}{x^2} \] Substitute this into the surface area expression: \[ S = x^2 + 4x \cdot \frac{V}{x^2} = x^2 + \frac{4V}{x} \]