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Answer:
Length = 2 ft
Width = 2 ft
Height = 5 ft
Step-by-step explanation:
Let the square base of the box has one side = x ft
Therefore, area of the base = x² ft²
Cost of the material to prepare the base = $0.35 per square feet
Cost to prepare the base = $0.35x²
Let the height of the box = y ft
Then the volume of the box = x²y ft³ = 20
-----(1)
Cost of the material for the sides = $0.10 per square feet
Area of the sides = 4xy
Cost to prepare the sides of the box = $0.10 × 4xy
= $0.40xy
Cost of the material to prepare the top = $0.15 per square feet
Cost to prepare the top = $0.15x²
Total cost of the box = 0.35x² + 0.40xy + 0.15x²
From equation (1),
Total cost
Now we take the derivative of C with respect to x and equate it to zero,
= 0
x = 2 ft.
From equation (1),
4y = 20
y = 5 ft
Therefore, Length and width of the box should be 2 ft and height of the box should be 5 ft for the minimum cost to construct the rectangular box.