Find the perimeter of a square with a side length of 212−−√

Question

. 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–√

Answer 1

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 \( 8\sqrt{53} \).

Answer 2

use the closest option given

Answer 3

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 \(8\sqrt{53}\).