Abox is to be made where the material for the sides and the lid cost $0.20 per square foot and the cost for the bottom is $0.50 per square foot. find the dimensions of a box with volume 5 cubic feet that has minimum cost.

Let x represent the edge dimension of the square base in feet. Then the area of the base or lid is x², and the height of the box is 5/x². Total cost is then ...

... cost = cost of lid + 4×(cost of one side) + cost of bottom

... cost = 0.2x² + 4×(0.2x(5/x²)) + 0.5x²

... cost = 0.7x² + 4/x

We can differentiate this with respect to x and set that derivative to zero. This will give us the x-value at which the cost is a minimum.

... d(cost)/dx = 1.4x -4/x² = 0

... x³ = 4/1.4

... x = ∛(20/7) ≈ 1.419

Then the height of the box is

... height = 5/x² ≈ 5/2.01351 ≈ 2.483

The box is 1.491 ft square and 2.483 ft high.

The average cost of the top and bottom is (0.2 +0.5)/2 = 0.35 per square foot. The overall cost will be minimized when the average cost of each face is the same. That is, the areas of the lateral faces will be equal and must be 0.35/0.2 = 1.75 times the area of the top or bottom to make the average cost of a lateral face equal to that of the top or bottom. Hence, the base will be square, and the box will be 1.75 times as high as it is wide or deep. The base edge dimension is then ∛(5/1.75) ≈ 1.419 ft.

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