Answer :
To determine which side length represents a square with an area of 450 square inches, let's analyze each option step-by-step.
### Square Side Lengths and Their Calculated Areas
1. Option 1: [tex]\( 15 \sqrt{2} \)[/tex] inches
- Calculate the area of the square with side length [tex]\( 15 \sqrt{2} \)[/tex]:
[tex]\[ \text{Area} = (15 \sqrt{2})^2 = 15^2 \times (\sqrt{2})^2 = 225 \times 2 = 450 \text{ square inches} \][/tex]
- Therefore, the area is 450 square inches.
2. Option 2: [tex]\( 15 \sqrt{3} \)[/tex] inches
- Calculate the area of the square with side length [tex]\( 15 \sqrt{3} \)[/tex]:
[tex]\[ \text{Area} = (15 \sqrt{3})^2 = 15^2 \times (\sqrt{3})^2 = 225 \times 3 = 675 \text{ square inches} \][/tex]
- Therefore, the area is 675 square inches.
3. Option 3: 112.5 inches
- Calculate the area of the square with a side length of 112.5 inches:
[tex]\[ \text{Area} = 112.5^2 = 12656.25 \text{ square inches} \][/tex]
- Therefore, the area is not even close to 450 square inches.
4. Option 4: 115.5 inches
- Calculate the area of the square with a side length of 115.5 inches:
[tex]\[ \text{Area} = 115.5^2 = 13340.25 \text{ square inches} \][/tex]
- Therefore, the area is not even close to 450 square inches.
### Conclusion
Only the first option, [tex]\( 15 \sqrt{2} \)[/tex] inches, results in a square with an area of 450 square inches. The correct side length of the square is:
[tex]\[ 15 \sqrt{2} \text{ inches} \][/tex]
### Square Side Lengths and Their Calculated Areas
1. Option 1: [tex]\( 15 \sqrt{2} \)[/tex] inches
- Calculate the area of the square with side length [tex]\( 15 \sqrt{2} \)[/tex]:
[tex]\[ \text{Area} = (15 \sqrt{2})^2 = 15^2 \times (\sqrt{2})^2 = 225 \times 2 = 450 \text{ square inches} \][/tex]
- Therefore, the area is 450 square inches.
2. Option 2: [tex]\( 15 \sqrt{3} \)[/tex] inches
- Calculate the area of the square with side length [tex]\( 15 \sqrt{3} \)[/tex]:
[tex]\[ \text{Area} = (15 \sqrt{3})^2 = 15^2 \times (\sqrt{3})^2 = 225 \times 3 = 675 \text{ square inches} \][/tex]
- Therefore, the area is 675 square inches.
3. Option 3: 112.5 inches
- Calculate the area of the square with a side length of 112.5 inches:
[tex]\[ \text{Area} = 112.5^2 = 12656.25 \text{ square inches} \][/tex]
- Therefore, the area is not even close to 450 square inches.
4. Option 4: 115.5 inches
- Calculate the area of the square with a side length of 115.5 inches:
[tex]\[ \text{Area} = 115.5^2 = 13340.25 \text{ square inches} \][/tex]
- Therefore, the area is not even close to 450 square inches.
### Conclusion
Only the first option, [tex]\( 15 \sqrt{2} \)[/tex] inches, results in a square with an area of 450 square inches. The correct side length of the square is:
[tex]\[ 15 \sqrt{2} \text{ inches} \][/tex]
