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
