Question

a box with a square base and open top must have a volume of 32,000 cm^2 . find the dimensions of the box that will minimize the amount of marerial used to make the box.

Answer 1

solution

The Volume of a box with a square base x by x cm and height h cm is V = x²y

Let x = length of a side of the base

y = height

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

Then, x²y = 32000, so y = 32000 / x²

Let S = surface area = area of base + total area of the 4 sides

The surface area of the box described is S = x² + 4xy

= x² + 4xy = x² + 4x(32000 / x²)

= x² + 128,000 / x, x > 0

Minimize S: S' = 2x - 128,000 / x²

= (2x³ - 128,000) / x²

S' = 0 when 2x³ = 128,000

x³ = 64,000

x = 40

When 0 < x < 40, S' < 0. So, S is decreasing.

When x > 40, S' > 0, so S is increasing.

Therefore, surface area is minimized when x = length of a side of the base= 40 cm

and y = height = 32000 / 40² = 20 cm.