A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

Question

side of base = ___cm

height = ____cm

Answer 1

Let x = length of a side of the base and y = height.

Volume = V = x²y = 4000. So, y = 4000 / x².

Surface area = S = x² + 4xy = x² + 4x(4000/x²) = x² + 16000x^-1

S' = 2x - 16000 / x² = (2x³ - 16000) / x²

S' = 0 when 2x³ = 16000. x³ = 8000

x = (8000)^1/3 = 20 and h = 4000 / (20)² = 10

The base is a square with side length 20 cm and the height is 10 cm.

Answer 2

The box should be a cube...but you should set up a simple equation using

V=s²h and S=s²+4sh where V is volume, s is the side of the square, h is the height of the box and S is the surface area of the box.

Set the derivative of the S in terms of either s or h=0 and solve.