Final answer:
The dimensions of the box that requires the least amount of cardboard, for a volume of 4 m³, are Length: 2 m, Width: 2 m, Height: 1 m, as these are closest to a cube which has the minimal surface area for any given volume.
Step-by-step explanation:
The student is asking for the dimensions of a lidless cardboard box that has a volume of 4 m³ and requires the least amount of cardboard. To solve this, we must use the concept of minimizing the surface area for a given volume. The volume of a box is calculated by multiplying its length (L), width (W), and height (H) together. We are given that L × W × H = 4 m³. To minimize the amount of cardboard used, which corresponds to the surface area, we are looking for equal or nearly equal dimensions. This is because a cube has the smallest surface area for a given volume.
The surface area for a box with no lid is calculated as SA = L×W + 2(L×H + W×H). Plugging in the given dimensions:
- For option A: SA = 2×2 + 2(2×1 + 2×1) = 4 + 8 = 12 m²
- For option B: SA = 4×1 + 2(4×1 + 1×1) = 4 + 14 = 18 m²
- For option C: SA = 1×2 + 2(1×2 + 2×2) = 2 + 10 = 12 m²
- For option D: SA = 2×1 + 2(2×2 + 1×2) = 2 + 12 = 14 m²
Options A and C have the smallest surface areas of 12 m², but since we want the dimensions that give the least amount of cardboard, and we have two options with the same surface area, we should consider which shape is closer to a cube. Option A, with dimensions 2m x 2m x 1m, is closer to a cube shape than option C, and thus it is the size of the box that will require the least amount of cardboard to make.