9514 1404 393
Answer:
3 ft square base, 7 feet high
Step-by-step 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).
