Answer :
To determine the ratio of the side lengths of two similar solids given the ratio of their surface areas is \(16:144\), follow these steps:
### Step 1: Understand the Surface Area-to-Side Length Relationship
For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding side lengths. This is because the surface area of a solid depends on the square of its linear dimensions.
### Step 2: Calculate the Side Length Ratios
Given the ratio of the surface areas is \(16:144\), we need to find the square roots of these numbers to determine the ratio of the side lengths.
#### Surface Area Ratio:
[tex]\[ \frac{16}{144} \][/tex]
#### Corresponding Side Length Ratios:
Taking the square root of each part of the ratio:
[tex]\[ \sqrt{16} = 4 \][/tex]
[tex]\[ \sqrt{144} = 12 \][/tex]
Thus, the ratio of the side lengths is:
[tex]\[ \frac{4}{12} \][/tex]
### Step 3: Simplify the Ratio
To simplify the ratio \( \frac{4}{12} \):
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
### Step 4: Interpret the Simplified Ratio
Thus, the ratio of the corresponding side lengths of the two similar solids is \(1:3\).
### Step 5: Match with the Given Options
Out of the given options, the correctly simplified ratio \(1:3\) matches none of the provided options directly. However, the closest correct mathematical interpretation we derived matches with option:
[tex]\[ \boxed{None of the above} \][/tex]
However, based on common understanding of the given answer choices, none of them correctly represent the simplified side length ratio \(1:3\). The precise answer is:
[tex]\[ 1:3 \][/tex]
Therefore, the ratio of their corresponding side lengths is [tex]\(1:3\)[/tex].
### Step 1: Understand the Surface Area-to-Side Length Relationship
For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding side lengths. This is because the surface area of a solid depends on the square of its linear dimensions.
### Step 2: Calculate the Side Length Ratios
Given the ratio of the surface areas is \(16:144\), we need to find the square roots of these numbers to determine the ratio of the side lengths.
#### Surface Area Ratio:
[tex]\[ \frac{16}{144} \][/tex]
#### Corresponding Side Length Ratios:
Taking the square root of each part of the ratio:
[tex]\[ \sqrt{16} = 4 \][/tex]
[tex]\[ \sqrt{144} = 12 \][/tex]
Thus, the ratio of the side lengths is:
[tex]\[ \frac{4}{12} \][/tex]
### Step 3: Simplify the Ratio
To simplify the ratio \( \frac{4}{12} \):
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
### Step 4: Interpret the Simplified Ratio
Thus, the ratio of the corresponding side lengths of the two similar solids is \(1:3\).
### Step 5: Match with the Given Options
Out of the given options, the correctly simplified ratio \(1:3\) matches none of the provided options directly. However, the closest correct mathematical interpretation we derived matches with option:
[tex]\[ \boxed{None of the above} \][/tex]
However, based on common understanding of the given answer choices, none of them correctly represent the simplified side length ratio \(1:3\). The precise answer is:
[tex]\[ 1:3 \][/tex]
Therefore, the ratio of their corresponding side lengths is [tex]\(1:3\)[/tex].
