Final answer:
For a closed rectangular box with a volume of 44 cm³, the edges giving the minimum surface area would be those of a cube. The cube root of the volume gives the length of each edge, which is approximately 3.530 cm. Therefore, all edges should be equal to minimize surface area.
Step-by-step explanation:
The question is asking for the lengths of the edges of a closed rectangular box with a given volume that would result in the minimum surface area. To solve this problem, we can use the properties of geometric optimization.
For a box with dimensions length (l), width (w), and height (h), the volume (V) is given by:
V = l × w × h
And the surface area (SA) is:
Since the volume is fixed at 44 cm³, we want to minimize SA. The box with the minimum surface area for a given volume is a cube, where all three dimensions are equal. Thus:
l = w = h = ∛(V)
Calculating the cube root of 44 cm³ gives the length of each edge for the cube with the minimum surface area:
l = w = h ≈ 3.530 cm
Therefore, to minimize the surface area, the edges of the rectangular box should be approximately 3.530 cm each.