Answer:
The smallest volume and surface area for each of the boxes of 12 can vary depending on the shape of the box. Let's consider a few different shapes and calculate their volumes and surface areas:
1. Cube:
- The volume of a cube is given by V = s^3, where s is the length of one side.
- If we want the smallest volume possible, we can choose s = 2 (since 2^3 = 8 is the smallest cube volume greater than 12).
- So, the volume of the cube would be V = 2^3 = 8 cubic units.
- The surface area of a cube is given by A = 6s^2.
- Therefore, the surface area of the cube would be A = 6(2^2) = 24 square units.
2. Rectangular prism:
- Let's consider a rectangular prism with dimensions a, b, and c.
- The volume of a rectangular prism is given by V = abc.
- To find the smallest volume, we need to find the factors of 12 and select the smallest combination of dimensions.
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- For example, if we choose a = 1, b = 2, and c = 6, then the volume would be V = 1(2)(6) = 12 cubic units.
- The surface area of a rectangular prism is given by A = 2(ab + ac + bc).
- So, the surface area of this rectangular prism would be A = 2(1(2) + 1(6) + 2(6)) = 32 square units.
3. Cylinder:
- Let's consider a cylinder with radius r and height h.
- The volume of a cylinder is given by V = πr^2h.
- Again, to find the smallest volume, we need to find the combination of dimensions that gives us a volume greater than 12.
- For example, if we choose r = 1 and h = 12, then the volume would be V = π(1^2)(12) = 12π cubic units (approximately 37.7 cubic units).
- The surface area of a cylinder is given by A = 2πr^2 + 2πrh.
- So, the surface area of this cylinder would be A = 2π(1^2) + 2π(1)(12) = 26π square units (approximately 81.7 square units).
Now, let's consider which of these shapes would require the smallest box to fit 12 cartoons in terms of both volume and surface area:
- The cube has the smallest volume (8 cubic units) and the smallest surface area (24 square units) among the three shapes we considered.
- Therefore, the cube would require the smallest box to fit 12 cartoons in terms of both volume and surface area.
It's important to note that there may be other shapes or combinations of dimensions that could yield smaller volumes or surface areas, but the examples provided above give you an idea of the calculations involved.
Explanation: