Answer:
- length: 2 ft
- width: 2 ft
- height: 10 ft
Explanation:
The total cost of the top and bottom is $0.36 + 0.14 = $0.50 per square foot.
The total cost of a pair of opposite sides is $0.05 +0.05 = $0.10 per square foot.
A minimum-cost box will have the cost of any pair of opposite sides be the same. Here, that means the box will have a side area that is 5 times the area of the top or bottom.
Since the base is square, that means the box is the shape of 5 cubes stacked one on the other. Each of those would be 8 ft³, so would have edge dimensions of ∛8 = 2 feet. The height is 5 times that, or 10 ft.
The box is 2 feet square by 10 feet high:
- length: 2 ft
- width: 2 ft
- height: 10 ft
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If you feel the need to write an equation, you can let x represent the edge length of the base. Then the cost of the top and bottom will be ...
top&bottom cost = 0.50·x²
The height of the box is 40/x², so the cost of the four sides will be ...
side cost = (0.05)(4x)(40/x²) = 8/x
This is minimized when the derivative of the sum of these costs is zero:
cost = top&bottom cost + side cost
cost = 0.50x² + 8/x
d(cost)/dx = 1.00x -8/x² = 0
Multiplying by x², we get ...
x³ -8 = 0
x = ∛8 = 2 . . . . . . as above
height = 40/x² = 40/4 = 10 . . . . . as above