Answer :
To determine the ratio of the side lengths of two similar solids given the ratio of their surface areas, we will proceed as follows:
1. Understand the Relationship:
- The ratio of the surface areas of similar solids is given as \( \frac{16}{144} \).
- For similar solids, the ratio of the surface areas is the square of the ratio of their corresponding side lengths. This means if \( k \) is the ratio of the side lengths, then \( k^2 \) is the ratio of the surface areas.
2. Find the Ratio of Side Lengths:
- Given the surface area ratio \( \frac{16}{144} \), we need to find \( k \) such that \( k^2 = \frac{16}{144} \).
- To find \( k \), we take the square root of the ratio of the surface areas:
[tex]\[ k = \sqrt{\frac{16}{144}} \][/tex]
3. Simplify the Ratio:
- Calculate the square root of both the numerator and the denominator:
[tex]\[ \sqrt{16} = 4 \quad \text{and} \quad \sqrt{144} = 12 \][/tex]
- Therefore:
[tex]\[ k = \frac{4}{12} \][/tex]
4. Simplify the Fraction:
- Simplify \( \frac{4}{12} \) by dividing both the numerator and the denominator by 4:
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
5. Express the Ratio in the Simplest Form:
- The ratio of the corresponding side lengths is \( \frac{1}{3} \).
Given the answer choices:
- A. \( 1:96 \)
- B. \( \frac{16}{12}:12 \)
- C. \( 4:\frac{144}{4} \)
- D. \( 4:12 \)
The correct answer is option D, which is [tex]\( 4:12 \)[/tex]. This simplifies to [tex]\( 1:3 \)[/tex], matching the simplified ratio we found for the side lengths.
1. Understand the Relationship:
- The ratio of the surface areas of similar solids is given as \( \frac{16}{144} \).
- For similar solids, the ratio of the surface areas is the square of the ratio of their corresponding side lengths. This means if \( k \) is the ratio of the side lengths, then \( k^2 \) is the ratio of the surface areas.
2. Find the Ratio of Side Lengths:
- Given the surface area ratio \( \frac{16}{144} \), we need to find \( k \) such that \( k^2 = \frac{16}{144} \).
- To find \( k \), we take the square root of the ratio of the surface areas:
[tex]\[ k = \sqrt{\frac{16}{144}} \][/tex]
3. Simplify the Ratio:
- Calculate the square root of both the numerator and the denominator:
[tex]\[ \sqrt{16} = 4 \quad \text{and} \quad \sqrt{144} = 12 \][/tex]
- Therefore:
[tex]\[ k = \frac{4}{12} \][/tex]
4. Simplify the Fraction:
- Simplify \( \frac{4}{12} \) by dividing both the numerator and the denominator by 4:
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
5. Express the Ratio in the Simplest Form:
- The ratio of the corresponding side lengths is \( \frac{1}{3} \).
Given the answer choices:
- A. \( 1:96 \)
- B. \( \frac{16}{12}:12 \)
- C. \( 4:\frac{144}{4} \)
- D. \( 4:12 \)
The correct answer is option D, which is [tex]\( 4:12 \)[/tex]. This simplifies to [tex]\( 1:3 \)[/tex], matching the simplified ratio we found for the side lengths.
