A storage box with a square base must have a volume of 70 cubic centimeters. The top and bottom cost $0.20 per square centimeter and the sides cost $0.10 per square centimeter.

Question

A storage box with a square base must have a volume of 70 cubic centimeters. The top and bottom cost $0.20 per square centimeter and the sides cost $0.10 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

x=

y=

Answer 1

Since x is representing the length of sides and y is representing the height, we know that the volume is

V = x²y = 70

We can also determine the cost of the surface area of the box. We know the top and bottom are the same size (x²) and all 4 sides are the same size (xy) so:

C = (0.20)(2x²) + (0.10)(4xy)

We have to determine where the cost will be minimized so when C' = 0. Let's first use the volume equation to solve for one of the variables and plug it in:

y = 70/x²

Now substitute and solve:

C = (0.20)(2x²) + (0.10)(4x(70/x²))

C = 0.4x² + 28/x

C' = 0.8x - 28/x² = 0

0.8x = 28/x²

0.8x³ = 28

x³ = 28/0.8 = 35

x = 3.27 cm

Now we can easily solve for y:

y = 70/x² = 70/(3.27²) = 6.54 cm