Question

You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 500cm3. Determine the dimensions of the bin that will minimize the surface area, where x is the length of each side of the base and y is the height of the box. Enter an exact answer.

Answer 1

x = 10 cm, y = 5 cm gives a minimum area of 300 cm^2.

Explanation:

V= x^2y = 500

Surface area A = x^2 + 4xy.

From the first equation y = 500/x^2

So substituting for y in the equation for the surface area:

A = x^2 + 4x * 500/x^2

A = x^2 + 2000/x

Finding the derivative:

dA/dx = 2x - 2000x^-2

dA/dx = 2x - 2000/x^2

This = 0 for a minimum/maximum value of A, so

2x - 2000/x^2 = 0

2x^3 - 2000 = 0

x^3 = 2000/ 2 = 1000

x = 10

Second derivative is 2 + 4000/x^3

when x = 10 this is positive so x = 10 gives a minimum value of A.

So y = 500/x^2

= 500/100

= 5.