Answer :
To determine which dimensions could be that of a right square prism with a volume of 360 cubic units, we need to check if the product of length, width, and height for each given option equals 360.
### Option 1: 3 by 3 by 40
1. Calculate the volume:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} = 3 \times 3 \times 40 \][/tex]
2. Perform the multiplication:
[tex]\[ 3 \times 3 = 9 \quad \text{and then} \quad 9 \times 40 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(3 \times 3 \times 40\) is a valid option.
### Option 2: 4 by 4 by 20
1. Calculate the volume:
[tex]\[ V = 4 \times 4 \times 20 \][/tex]
2. Perform the multiplication:
[tex]\[ 4 \times 4 = 16 \quad \text{and then} \quad 16 \times 20 = 320 \][/tex]
3. The volume is 320 cubic units, which does not match the required volume.
So, \(4 \times 4 \times 20\) is not a valid option.
### Option 3: 5 by 5 by 14
1. Calculate the volume:
[tex]\[ V = 5 \times 5 \times 14 \][/tex]
2. Perform the multiplication:
[tex]\[ 5 \times 5 = 25 \quad \text{and then} \quad 25 \times 14 = 350 \][/tex]
3. The volume is 350 cubic units, which does not match the required volume.
So, \(5 \times 5 \times 14\) is not a valid option.
### Option 4: 2.5 by 12 by 12
1. Calculate the volume:
[tex]\[ V = 2.5 \times 12 \times 12 \][/tex]
2. Perform the multiplication:
[tex]\[ 2.5 \times 12 = 30 \quad \text{and then} \quad 30 \times 12 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(2.5 \times 12 \times 12\) is a valid option.
### Option 5: 3.6 by 10 by 10
1. Calculate the volume:
[tex]\[ V = 3.6 \times 10 \times 10 \][/tex]
2. Perform the multiplication:
[tex]\[ 3.6 \times 10 = 36 \quad \text{and then} \quad 36 \times 10 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(3.6 \times 10 \times 10\) is a valid option.
### Conclusion
The three valid dimensions for the right square prism with a volume of 360 cubic units are:
1. \(3 \times 3 \times 40\)
2. \(2.5 \times 12 \times 12\)
3. \(3.6 \times 10 \times 10\)
These options satisfy the given volume requirement.
### Option 1: 3 by 3 by 40
1. Calculate the volume:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} = 3 \times 3 \times 40 \][/tex]
2. Perform the multiplication:
[tex]\[ 3 \times 3 = 9 \quad \text{and then} \quad 9 \times 40 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(3 \times 3 \times 40\) is a valid option.
### Option 2: 4 by 4 by 20
1. Calculate the volume:
[tex]\[ V = 4 \times 4 \times 20 \][/tex]
2. Perform the multiplication:
[tex]\[ 4 \times 4 = 16 \quad \text{and then} \quad 16 \times 20 = 320 \][/tex]
3. The volume is 320 cubic units, which does not match the required volume.
So, \(4 \times 4 \times 20\) is not a valid option.
### Option 3: 5 by 5 by 14
1. Calculate the volume:
[tex]\[ V = 5 \times 5 \times 14 \][/tex]
2. Perform the multiplication:
[tex]\[ 5 \times 5 = 25 \quad \text{and then} \quad 25 \times 14 = 350 \][/tex]
3. The volume is 350 cubic units, which does not match the required volume.
So, \(5 \times 5 \times 14\) is not a valid option.
### Option 4: 2.5 by 12 by 12
1. Calculate the volume:
[tex]\[ V = 2.5 \times 12 \times 12 \][/tex]
2. Perform the multiplication:
[tex]\[ 2.5 \times 12 = 30 \quad \text{and then} \quad 30 \times 12 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(2.5 \times 12 \times 12\) is a valid option.
### Option 5: 3.6 by 10 by 10
1. Calculate the volume:
[tex]\[ V = 3.6 \times 10 \times 10 \][/tex]
2. Perform the multiplication:
[tex]\[ 3.6 \times 10 = 36 \quad \text{and then} \quad 36 \times 10 = 360 \][/tex]
3. The volume is 360 cubic units, which matches the required volume.
So, \(3.6 \times 10 \times 10\) is a valid option.
### Conclusion
The three valid dimensions for the right square prism with a volume of 360 cubic units are:
1. \(3 \times 3 \times 40\)
2. \(2.5 \times 12 \times 12\)
3. \(3.6 \times 10 \times 10\)
These options satisfy the given volume requirement.
