Final Answer:
To minimize the amount of material used, the dimensions of the box with a square base and open top that achieves a volume of 108,000 cubic units are a square base of 30^2 units and a height of 120 units.
Step-by-step explanation:
The problem involves optimization, specifically minimizing the surface area of the box, which is equivalent to minimizing the amount of material used. Let x represent the side length of the square base, and h represent the height of the box. The volume of the box is given by V = x^2 * h, and the surface area is S = x^2 + 4xh.
Since the volume is fixed at 108,000 cubic units, we have x^2 * h = 108,000. To minimize the surface area, we need to express h in terms of x using the volume constraint. Solving for h, we get h = 108,000 / x^2. Substituting this expression for h into the surface area formula, we have S = x^2 + 4x * 108,000 / x^2.
To find the critical points of S, we take the derivative, set it equal to zero, and solve for x. After finding the critical points, we can determine which one corresponds to the minimum surface area. In this case, the dimensions that minimize the amount of material used are a square base with side length x = 30 units and a height of h = 120 units, resulting in a volume of 108,000 cubic units. This configuration minimizes the surface area, fulfilling the optimization requirement.