To determine which volume could belong to a cube with a side length that is an integer, we need to check if any of the given volumes can be expressed in the form [tex]\( V = s^3 \)[/tex], where [tex]\( s \)[/tex] is an integer.
Let's examine each given volume:
1. 18 cubic inches:
- We need to find if there is an integer [tex]\( s \)[/tex] such that [tex]\( s^3 = 18 \)[/tex].
- Checking the cube roots of small integers:
- [tex]\( 1^3 = 1 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 3^3 = 27 \)[/tex]
- None of these cube values equal 18.
- Hence, 18 cubic inches cannot be represented as a perfect cube with an integer side length.
2. 36 cubic inches:
- We need to find if there is an integer [tex]\( s \)[/tex] such that [tex]\( s^3 = 36 \)[/tex].
- Checking the cube roots of small integers:
- [tex]\( 1^3 = 1 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 4^3 = 64 \)[/tex]
- None of these cube values equal 36.
- Hence, 36 cubic inches cannot be represented as a perfect cube with an integer side length.
3. 64 cubic inches:
- We need to find if there is an integer [tex]\( s \)[/tex] such that [tex]\( s^3 = 64 \)[/tex].
- Checking the cube roots of small integers:
- [tex]\( 1^3 = 1 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 4^3 = 64 \)[/tex]
- Here, we find that [tex]\( 4^3 = 64 \)[/tex].
- Thus, 64 cubic inches can be represented as a perfect cube with an integer side length, specifically [tex]\( s = 4 \)[/tex].
4. 100 cubic inches:
- We need to find if there is an integer [tex]\( s \)[/tex] such that [tex]\( s^3 = 100 \)[/tex].
- Checking the cube roots of small integers:
- [tex]\( 1^3 = 1 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 4^3 = 64 \)[/tex]
- [tex]\( 5^3 = 125 \)[/tex]
- None of these cube values equal 100.
- Hence, 100 cubic inches cannot be represented as a perfect cube with an integer side length.
In conclusion, the volume that can belong to a cube with a side length that is an integer is 64 cubic inches.
