A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.36/ft2, the material for the sides costs $0.05/ft2, and the material for the top costs $0.14/ft2, determine the dimensions of the box that can be constructed at minimum cost.length ftwidth ftheight ft

Answer:length: 2 ftwidth: 2 ftheight: 10 ftStep-by-step 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 ftwidth: 2 ftheight: 10 ft_____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·x²The height of the box is 40/x², so the cost of the four sides will be ...   side cost = (0.05)(4x)(40/x²) = 8/xThis 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 = 0Multiplying by x², we get ...   x³ -8 = 0   x = ∛8 = 2 . . . . . . as above   height = 40/x² = 40/4 = 10 . . . . . as above