Let the side length of the square base be x cm and the height of the box be h cm. Then the volume of the box is given by:
V = x^2 * h = 13,500 cm^3
We want to minimize the amount of material used, which is the surface area of the box. The surface area A is given by:
A = x^2 + 4xh
We can use the volume equation to solve for h in terms of x:
h = 13,500 / x^2
Substituting this into the surface area equation gives:
A = x^2 + 4x(13,500 / x^2) = x^2 + 54,000 / x
To minimize A, we take the derivative with respect to x and set it equal to zero:
dA/dx = 2x - 54,000 / x^2 = 0
Solving for x, we get:
x^3 = 27,000
x = 30
Substituting this back into the volume equation gives:
h = 13,500 / (30^2) = 15
Therefore, the dimensions of the box that minimize the amount of material used are 30 cm by 30 cm by 15 cm.