Final answer:
The dimensions of the edges that give the minimum surface area for a given volume of a closed rectangular box are those of a cube, where length, width, and height are equal. The edge length is the cube root of the volume.
Step-by-step explanation:
To determine the dimensions of a rectangular box with a given volume that has the minimum surface area, we need to consider that a box with equal sides—a cube—will have the smallest surface area for a given volume. Suppose we have a closed rectangular box with a specific volume V. To find the lengths of its edges that would result in the minimum surface area, we use the concept that for a constant volume, a cube has the minimum surface area compared to any other rectangular box shape.
Let's denote the lengths of the edges as l (length), w (width), and h (height). The volume of the box is given by V = l × w × h. The surface area S is given by S = 2lw + 2lh + 2wh. To minimize S while keeping V constant, we set each dimension equal: l = w = h, hence the shape is a cube. Therefore, for a volume V, the lengths of the edges, which are all equal, can be found by solving the equation l = ∛(V).