Arectangular box is to have a square base and a volume of 60 ft3. the material for the base costs 33¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 20¢/ft2. letting x denote the length of one side of the base, find a function in the variable x giving the cost (in dollars) of constructing the box.

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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