To determine how many identical cuboids of dimensions [tex]\(3 \, \text{cm} \times 5 \, \text{cm} \times 5 \, \text{cm}\)[/tex] are needed to form a perfect cube, follow these steps:
1. Calculate the volume of the cuboid:
- A cuboid’s volume is given by its length multiplied by its width and height.
- [tex]\[
\text{Volume of cuboid} = 3 \, \text{cm} \times 5 \, \text{cm} \times 5 \, \text{cm} = 75 \, \text{cm}^3
\][/tex]
2. Determine the side length of the smallest cube that can encompass the volume of one cuboid:
- We need to find the cube root of the cuboid's volume to determine the side length of the smallest cube that can contain this volume.
- [tex]\[
\text{Side length of cube} \approx \sqrt[3]{75} \approx 4.217
\][/tex]
- Since we need to form a perfect cube, this side length should be rounded to the nearest whole number.
- [tex]\[
\text{Rounded side length} = 4
\][/tex]
3. Calculate the volume of the smallest perfect cube:
- A cube's volume is given by the side length cubed.
- [tex]\[
\text{Volume of cube} = 4 \, \text{cm} \times 4 \, \text{cm} \times 4 \, \text{cm} = 64 \, \text{cm}^3
\][/tex]
4. Determine how many cuboids are required to form this cube:
- To find out how many cuboids can fit into this cube, divide the volume of the cube by the volume of the cuboid.
- [tex]\[
\text{Number of cuboids} = \frac{\text{Volume of cube}}{\text{Volume of cuboid}} = \frac{64 \, \text{cm}^3}{75 \, \text{cm}^3}
\][/tex]
5. Calculate the result:
- The ratio of the volumes [tex]\(\frac{64}{75}\)[/tex] is less than 1, indicating that a single cuboid does not fit into the smaller cube.
Thus, the calculation shows that it is not possible to fit the given cuboids into a perfect cube with the computed dimensions.
