Question

A box with a square base and open top must have a volume of 13,500 cm3. Find the dimensions of the box that minimize the amount of material used

Answer 1

Explanation:

Let the side of the square base be x

h be the height of the box

Volume V = x²h

13500 = x²h

h = 13500/x² ..... 1

Surface area = x² + 2xh + 2xh

Surface area S = x² + 4xh ...... 2

Substitute 1 into 2;

From 2; S = x² + 4xh

S = x² + 4x(13500/x²)

S = x² + 54000/x

To minimize the amount of material used; dS/dx = 0

dS/dx = 2x - 54000/x²

0 = 2x - 54000/x²

0 = 2x³ - 54000

2x³ = 54000

x³ = 27000

x = ∛27000

x = 30cm

Since V = x²h

13500 = 30²h

h = 13500/900

h = 15cm

Hence the dimensions of the box that minimize the amount of material used is 30cm by 30cm by 15cm