Question

An open box is constructed of 3500 cm2 of cardboard. The box is a cuboid, with height hcm and square base of side xcm. What is the value of x which maximises the volume of the box?

Answer 1

34.16 cm

Explanation:

side of square base = x

height = h

area, A = 3500 cm^2

Area = x² + 4xh = 3500

4 xh = 3500 - x²

h = (3500 - x²)/4x

Volume = Area of base x height

V = x² h

V = x² (3500 - x²)/4x

V = (3500 x - x³) / 4

Differentiate volume with respect to x

dV/dx = (3500 - 3x²) / 4

It is equal to zero for maxima and minima

3500 - 3x² = 0

x = 34.16 cm

Now differentiate again

d²V/dx² = 6x / 4

It is negative so the volume is maximum.

Thus, for x = 34.16 cm, the volume is maximum.