A rectangular box is to have a square base and a volume of 80 ft3. the material for the base costs 45¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 28¢/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 volume of a rectangular box is equal to the area of its base times the vertical height. Thus,

V = Ah = 80

Since the base is a square with side x, then the area is equal to x². Hence, we can express h in terms of x.

80 = x²h

h = 80/x²

Now, each lateral side of the rectangular box is in the shape of a rectangle with a length of h and a width of w. Hence, the equation for the total cost would be:

Total Cost = Cost per area of lower base + Cost per area of upper base + 4 (Cost per area of lateral side)

Total Cost = 0.45x² + 0.28x² + 4 (0.10h*x)

Total Cost = 0.73x² + 0.4hx

Since h = 80/x²,

Total Cost = 0.73x² + 0.4 (80/x²) (x)

Total Cost ($) = 0.73x² + 32/x