Question

You are asked to construct a rectangular crate with a square

base. The volume must be 12 cubic feet. The material for the bottom costs
$8 per square foot, the material for the top costs $2 per square foot, and the
material for the sides costs $4 per square foot. Find the dimensions of a
crate that yield a minimum cost. Show all work.

Answer 1

2.13 ft x 2.13 ft x 2.66 ft

Explanation:

If W is the width of the square base, and H is the height, then:

12 = W²H

The cost of the crate is:

C = 8W² + 2W² + 4(4WH)

C = 10W² + 16WH

Solve for H in the first equation:

H = 12 / W²

Substitute into the second equation:

C = 10W² + 16W (12 / W²)

C = 10W² + 192 / W

Take the derivative with respect to W.

dC/dW = 20W − 192 / W²

Set to 0 and solve:

0 = 20W − 192 / W²

20W = 192 / W²

20W³ = 192

5W³ = 48

W = ∛(48/5)

W ≈ 2.13 ft

Find H:

H = 12 / W²

H ≈ 2.66 ft

Answer 2

2.125 ft square by 2.657 ft high

Explanation:

Let x represent the side length in feet of the square base. Then the height is found from ...

V = Bh

V/B = h = 12/x²

The cost of material is the sum of costs of top, bottom, and sides. That is ...

cost = 8x² +2x² +4(4xh) = 10x² +192/x

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The cost is minimized when its derivative with respect to x is zero.

d(cost)/dx = 20x -192/x² = 0

x³ = 9.6 . . . . . . multiply by x²/20 and add 9.6

x = ∛9.6 ≈ 2.125 . . . . feet

h = 12/x² ≈ 2.657 . . . . feet

The dimensions of the crate are 2.125 feet square by 2.657 feet high.

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Comment on the answer

You may notice that the cost of top and bottom together is $10 per square foot. The cost of opposite sides together is $8 per square foot. That is, the ratio of costs is ...

base : sides = 10 : 8

You may also notice that the ratio of dimensions is ...

height : base side = 2.657 : 2.125 = 10 : 8

That is no accident. The dimensions of the lowest-cost crate are such that any pair of opposite sides costs the same.