Final answer:
To identify the box dimensions with maximum volume and a fixed surface area, calculus optimization techniques such as finding the derivative of the volume expression are used to determine the box dimensions that yield the greatest volume.
Step-by-step explanation:
To find the dimensions that produce a box with maximum volume, given a fixed surface area of 108 in², we need to employ optimization techniques from calculus, specifically using derivatives to find the maximum of a function. Assuming the dimensions of the box are x for width and length of the square base, and h for the height, the surface area S of an open box (with one face missing) is S = x^2 + 4xh. To maximize the volume V = x^2h, we set up the equation for the surface area and use it to express h in terms of x. Plugging this expression for h into the volume equation, and then taking the derivative of V with respect to x, allows us to find the critical points where the volume might be maximized. Solving this gives us the optimal dimensions of the box for maximum volume. For a given surface area of 108 in², the dimensions that will yield the maximum volume according to the calculus approach are not explicitly listed among options A to D, but by using the method described, one can find the exact dimensions that accomplish this.