Answer :
To determine which volumes could belong to a cube with a side length that is an integer, we need to check each given volume to see if it satisfies the cube volume formula \( V = s^3 \), where \( s \) is the side length of the cube.
Here are the steps to solve this:
1. Let's test 18 cubic inches:
[tex]\[ s = \sqrt[3]{18} \][/tex]
Calculating \( \sqrt[3]{18} \) does not yield an integer. So, 18 cubic inches is not the volume of a cube with an integer side length.
2. Let's test 36 cubic inches:
[tex]\[ s = \sqrt[3]{36} \][/tex]
Calculating \( \sqrt[3]{36} \) does not yield an integer. So, 36 cubic inches is not the volume of a cube with an integer side length.
3. Let's test 64 cubic inches:
[tex]\[ s = \sqrt[3]{64} = 4 \][/tex]
The cube root of 64 is 4, which is an integer. Therefore, 64 cubic inches is the volume of a cube with an integer side length (4 inches).
4. Let's test 100 cubic inches:
[tex]\[ s = \sqrt[3]{100} \][/tex]
Calculating \( \sqrt[3]{100} \) does not yield an integer. So, 100 cubic inches is not the volume of a cube with an integer side length.
After these calculations, we can see that the only volume that could belong to a cube with a side length that is an integer is 64 cubic inches.
Here are the steps to solve this:
1. Let's test 18 cubic inches:
[tex]\[ s = \sqrt[3]{18} \][/tex]
Calculating \( \sqrt[3]{18} \) does not yield an integer. So, 18 cubic inches is not the volume of a cube with an integer side length.
2. Let's test 36 cubic inches:
[tex]\[ s = \sqrt[3]{36} \][/tex]
Calculating \( \sqrt[3]{36} \) does not yield an integer. So, 36 cubic inches is not the volume of a cube with an integer side length.
3. Let's test 64 cubic inches:
[tex]\[ s = \sqrt[3]{64} = 4 \][/tex]
The cube root of 64 is 4, which is an integer. Therefore, 64 cubic inches is the volume of a cube with an integer side length (4 inches).
4. Let's test 100 cubic inches:
[tex]\[ s = \sqrt[3]{100} \][/tex]
Calculating \( \sqrt[3]{100} \) does not yield an integer. So, 100 cubic inches is not the volume of a cube with an integer side length.
After these calculations, we can see that the only volume that could belong to a cube with a side length that is an integer is 64 cubic inches.
