A closed box is constructed of 4200cm2 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

Volume = x²h

Surface area is 2x² + 4xh = 4200

So h = (4200 - 2x²) / 4x

V = x² (4200-2x²⁾ / 4x = x (4200-2x^2)//4

To maximise V, solve dV/dx = 0

ie 4200/4 - 2/4. 3x² = 0 or 4200 - 6x² = 0 , giving x = 26.5, h = 26.5, and volume 18609.6 cm³

Mike