Final answer:
To find the height that would maximize the volume of the box made from the given sheet of cardboard, we need to consider the dimensions of the cardboard and the size of the squares cut from each corner. By applying the volume formula and finding the critical points, we determine that the height that would maximize the volume is 3 ft.
Step-by-step explanation:
To find the height that would maximize the volume of the box made from the given sheet of cardboard, we need to consider the dimensions of the cardboard and the size of the squares cut from each corner. The formula for the volume of a rectangular box is V = lwh, where l is the length, w is the width, and h is the height. Given that the cardboard is 12 ft. by 18 ft., and we cut equal-sized squares from each corner, let's assume the side length of the squares cut is x ft. Therefore, the length of the box would be (12 - 2x) ft., the width would be (18 - 2x) ft., and the height would be x ft.
The volume formula becomes: V = (12 - 2x)(18 - 2x)(x). To maximize the volume, we need to find the value of x that gives the maximum result when substituted into the volume formula. This can be done by expanding the formula, differentiating with respect to x, and finding the critical points. After some algebraic manipulation, we find that the height that would maximize the volume is x = 3 ft.
Therefore, the correct answer is B. 3.06 feet.