Question

A rectangular box is to have a square base and a volume of 63 ft3. If the material for the base costs $0.22/ft2, the material for the sides costs $0.09/ft2, and the material for the top costs $0.20/ft2, determine the dimensions (in ft) of the box that can be constructed at minimum cost.

Answer 1

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Answer:

3 ft square base, 7 feet high

Explanation:

For minimum cost, each pair of opposite sides costs the same as any other pair of opposite sides.

If s is the edge length of the base, then the areas of the top and bottom are s² and their total cost is ...

(s²)(0.22 +0.20) = 0.42s² . . . . cost of top and bottom

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For a volume of 63 ft³, the height of the side of the box is ...

63/s² = h

and the area of one side of the box is ...

hs = 63/s

A pair of opposite sides will have a cost of ...

(63/s)(0.09 +0.09) = 11.34/s . . . . cost of a pair of opposite sides

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We want these two costs to be the same, so we have ...

0.42s² = 11.34/s

s³ = 11.34/0.42 = 27

s = ∛27 = 3 . . . . feet

h = 63/s² = 7 . . . . feet

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The box dimensions are 3 feet square by 7 feet tall.

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Additional comment

Each pair of opposite sides will cost $3.78, so the total cost of the box is $11.34.

For the non-believers, we have shown a graph of the total cost of the box as a function of the base dimension (s). It has a minimum at $11.34 when s=3. You can find that by differentiating the cost function and and finding the zero(s) of that (or by reading the values from the graph).