The function in the variable x that gives the cost of constructing the box is;
C(x) = $0.53x² + $24/x
The base of the box is square.
If the side of the square base is x, then area of base is;
A_base = x²
Cost of base per sq.ft = 33 ¢/ft² = 0.33 $/ft²
Thus, cost of base = $ 0.33x²
Since the bottom is square, then it means that the top too will be square. Thus; Area of top = x²
Cost of top per sq.ft = 20 ¢/ft² = 0.2 $/ft²
Thus, cost of top = $0.2x²
Now, if the height of the box is y and we are told that the volume of the box is 60 ft³;
Since formula for box volume is V = length x width x height
Then; 60 = x²y
y = 60/x²
Thus;
Area of one side = x(60/x²)
Area of one side = 60/x
Area of four sides = 4(60/x) = 240/x
We are given that cost for the sides is 10 ¢/ft2 = 0.1 $/ft²
Thus, cost of sides = 0.1 × 240/x
Cost of sides = $24/x
The total cost of constructing the box is;
Total cost = cost of base + cost of top + cost of sides
Total cost = $ 0.33x² + $0.2x² + $24/x
Total Cost = $0.53x² + $24/x
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