Answer :
To determine which volumes are perfect cubes, you want to identify the numbers that can be expressed as the cube of an integer. Let's check each given volume:
1. 0.1 in.³: The cube root of 0.1 is a small, non-integer number. Therefore, 0.1 is not a perfect cube.
2. 0.4 in.³: The cube root of 0.4 also results in a non-integer number. Hence, 0.4 is not a perfect cube.
3. 8 in.³: The cube root of 8 is 2, since 2 × 2 × 2 = 8. Thus, 8 is a perfect cube.
4. 12 in.³: The cube root of 12 is not an integer since it falls between 2 and 3. Therefore, 12 is not a perfect cube.
5. 25 in.³: The cube root of 25 is a non-integer because it's between 2 and 3. So, 25 is not a perfect cube.
6. 27 in.³: The cube root of 27 is 3, since 3 × 3 × 3 = 27. So, 27 is a perfect cube.
7. 64 in.³: The cube root of 64 is 4, since 4 × 4 × 4 = 64. Thus, 64 is a perfect cube.
In conclusion, the volumes that represent perfect cubes are 8 in.³, 27 in.³, and 64 in.³.
1. 0.1 in.³: The cube root of 0.1 is a small, non-integer number. Therefore, 0.1 is not a perfect cube.
2. 0.4 in.³: The cube root of 0.4 also results in a non-integer number. Hence, 0.4 is not a perfect cube.
3. 8 in.³: The cube root of 8 is 2, since 2 × 2 × 2 = 8. Thus, 8 is a perfect cube.
4. 12 in.³: The cube root of 12 is not an integer since it falls between 2 and 3. Therefore, 12 is not a perfect cube.
5. 25 in.³: The cube root of 25 is a non-integer because it's between 2 and 3. So, 25 is not a perfect cube.
6. 27 in.³: The cube root of 27 is 3, since 3 × 3 × 3 = 27. So, 27 is a perfect cube.
7. 64 in.³: The cube root of 64 is 4, since 4 × 4 × 4 = 64. Thus, 64 is a perfect cube.
In conclusion, the volumes that represent perfect cubes are 8 in.³, 27 in.³, and 64 in.³.
