A rectangular box is to have a square base and a volume of 30 ft3. The material for the base costs 33¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 24¢/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.

Let each side of the base be x ft, and its height be h ft

Volume = x^2 h
30/x^2 = h

cost of base = 33x^2
cost of sides = 10(4)(xh) = 40x(30/x^2 = 1200/x
cost of top = 24x^2

cost = 57x^2 + 1200/x

To find the cost of constructing the box, we need to determine the dimensions of the box and calculate the cost of each component.

Let's denote the length of one side of the base as x. Since the base is square, the base dimensions would be x by x, resulting in an area of x^2 ft^2.

To find the height of the box, we can use the formula for volume: Volume = Length × Width × Height. In this case, the volume is given as 30 ft^3, and the base dimensions are x by x. So, the height of the box would be 30 / (x^2) ft.

Now, we can calculate the cost of constructing the box. There are three components: the base, the sides, and the top.

1. The cost of the base:
- The cost is given as 33¢/ft^2.
- The area of the base is x^2 ft^2.
- Hence, the cost of the base is (33¢/ft^2) * (x^2 ft^2).

2. The cost of the sides:
- The cost is given as 10¢/ft^2.
- There are four sides, each with the height of the box and the length equal to the base side length x.
- So, the total cost of the sides is 4 * (10¢/ft^2) * (x ft) * (30 / (x^2) ft).

3. The cost of the top:
- The cost is given as 24¢/ft^2.
- The area of the top is x^2 ft^2.
- Hence, the cost of the top is (24¢/ft^2) * (x^2 ft^2).

Thus, the total cost of constructing the box is the sum of the costs of the base, sides, and top:

Cost(x) = (33¢/ft^2) * (x^2 ft^2) + 4 * (10¢/ft^2) * (x ft) * (30 / (x^2) ft) + (24¢/ft^2) * (x^2 ft^2).

Simplifying this equation will provide a function in the variable x that gives the cost in dollars of constructing the box.