Question
Find the perimeter of a square with a side length of 212−−√
. Your answer must be in simplest radical form.(1 point)
Responses
163–√
16 square root of 3
812−−√
8 square root of 12
848−−√
8 square root of 48
323–√
. Your answer must be in simplest radical form.(1 point)
Responses
163–√
16 square root of 3
812−−√
8 square root of 12
848−−√
8 square root of 48
323–√
Answers
Answered by
GPT-4o mini
To find the perimeter of a square, we use the formula:
\[
\text{Perimeter} = 4 \times \text{side length}
\]
Given the side length is \( \sqrt{212} \), we can substitute this value into the formula:
\[
\text{Perimeter} = 4 \times \sqrt{212}
\]
Next, we can simplify \( \sqrt{212} \):
\[
\sqrt{212} = \sqrt{4 \times 53} = \sqrt{4} \times \sqrt{53} = 2\sqrt{53}
\]
Now substituting back into the perimeter formula:
\[
\text{Perimeter} = 4 \times 2\sqrt{53} = 8\sqrt{53}
\]
Thus, the perimeter of the square is:
\[
\boxed{8\sqrt{53}}
\]
Please note that this answer does not match any of the provided options, which may indicate a misunderstanding. However, based on the calculations, the correct simplest radical form for the perimeter given the side length is indeed \( 8\sqrt{53} \).
\[
\text{Perimeter} = 4 \times \text{side length}
\]
Given the side length is \( \sqrt{212} \), we can substitute this value into the formula:
\[
\text{Perimeter} = 4 \times \sqrt{212}
\]
Next, we can simplify \( \sqrt{212} \):
\[
\sqrt{212} = \sqrt{4 \times 53} = \sqrt{4} \times \sqrt{53} = 2\sqrt{53}
\]
Now substituting back into the perimeter formula:
\[
\text{Perimeter} = 4 \times 2\sqrt{53} = 8\sqrt{53}
\]
Thus, the perimeter of the square is:
\[
\boxed{8\sqrt{53}}
\]
Please note that this answer does not match any of the provided options, which may indicate a misunderstanding. However, based on the calculations, the correct simplest radical form for the perimeter given the side length is indeed \( 8\sqrt{53} \).
Answer
use the closest option given
Answered by
GPT-4o mini
To match the perimeter calculation \(8\sqrt{53}\) to the closest option given, let's evaluate each one:
1. **Option: \(8\sqrt{12}\)**
- Simplifying \(8\sqrt{12}\):
\[
8\sqrt{12} = 8 \times \sqrt{4 \times 3} = 8 \times 2\sqrt{3} = 16\sqrt{3}
\]
2. **Option: \(8\sqrt{48}\)**
- Simplifying \(8\sqrt{48}\):
\[
8\sqrt{48} = 8 \times \sqrt{16 \times 3} = 8 \times 4\sqrt{3} = 32\sqrt{3}
\]
3. **Option: \(\sqrt{323}\)**
- This does not directly relate to our calculation since it’s not in the form of \(a\sqrt{b}\).
4. **Option: \(16\sqrt{3}\)** (Already simplified)
- As shown above, it simplifies to \(16\sqrt{3}\).
Comparing \(8\sqrt{53}\):
To approximate \( \sqrt{53} \):
- \(\sqrt{53} \approx 7.28\)
- Thus, \(8\sqrt{53} \approx 8 \times 7.28 \approx 58.24\).
We don't have an exact match, but we can quickly estimate:
- \(16\sqrt{3} \approx 16 \times 1.73 \approx 27.68\)
- \(32\sqrt{3} \approx 32 \times 1.73 \approx 55.36\)
Given the approximations, \(8\sqrt{48}\) aligns closest with \(8\sqrt{53}\) when considering the options provided. Therefore, the closest option to the computed perimeter of a square with a side length of \( \sqrt{212} \) is:
\[
\boxed{8\sqrt{48}}
\]
as it compares favorably in magnitude to \(8\sqrt{53}\).
1. **Option: \(8\sqrt{12}\)**
- Simplifying \(8\sqrt{12}\):
\[
8\sqrt{12} = 8 \times \sqrt{4 \times 3} = 8 \times 2\sqrt{3} = 16\sqrt{3}
\]
2. **Option: \(8\sqrt{48}\)**
- Simplifying \(8\sqrt{48}\):
\[
8\sqrt{48} = 8 \times \sqrt{16 \times 3} = 8 \times 4\sqrt{3} = 32\sqrt{3}
\]
3. **Option: \(\sqrt{323}\)**
- This does not directly relate to our calculation since it’s not in the form of \(a\sqrt{b}\).
4. **Option: \(16\sqrt{3}\)** (Already simplified)
- As shown above, it simplifies to \(16\sqrt{3}\).
Comparing \(8\sqrt{53}\):
To approximate \( \sqrt{53} \):
- \(\sqrt{53} \approx 7.28\)
- Thus, \(8\sqrt{53} \approx 8 \times 7.28 \approx 58.24\).
We don't have an exact match, but we can quickly estimate:
- \(16\sqrt{3} \approx 16 \times 1.73 \approx 27.68\)
- \(32\sqrt{3} \approx 32 \times 1.73 \approx 55.36\)
Given the approximations, \(8\sqrt{48}\) aligns closest with \(8\sqrt{53}\) when considering the options provided. Therefore, the closest option to the computed perimeter of a square with a side length of \( \sqrt{212} \) is:
\[
\boxed{8\sqrt{48}}
\]
as it compares favorably in magnitude to \(8\sqrt{53}\).