Arturo builds a storage box with a square base. The box should have a volume of 80 cubic centimeters.

Question

Arturo builds a storage box with a square base. The box should have a volume of 80 cubic centimeters. The material used for the top and bottom of the box (the square bases) cost $0.21 per square centimeter, while the lateral faces cost $0.31 per square centimeter. What dimensions will minimize the cost of the box?

Answer 1

Let x be the dimensions of the square base and y be the height.

Volume, V = x²y = 80 --> y = 80/x²

Cost=0.21(2x²+0.32(4xy)

C(x) = 0.42x²+1.24x(80/x²) = 0.42x²+99.2/x

To find the minimum, take the derivative, set that to zero and solve for x.

C'(x) = 0.84x - 99.2/x²

0.84x³ = 99.2

x = (99.2/0.84)^1/3 = 4.906

y = 80/x² = 3.3235

The square base dimension is 4.906 cm and the height is 3.3235 cm.

Answer 2

Suppose base is b x b and height h

Then volume = b² h = 80

Material cost C = 2 b² x 0.21 + 4bh x 0.31 = 0.42b² + 1.24 bh

h = 80/b²

C = 0.42b² + (1.24b x 80/b²) = 0.42b² + 99.2 / b

We need dC/dx = 0

dC/dx = 0.84b - 99.2/b²

dC/dx = 0 when b³ = (99.2 / 0.84) giving b = 4.906 and h = 3.324