A stretched wire of length `114cm` is divided into three segments whose frequencies are in the ratio `1 : 3 : 4`, the lengths of the segments must be in the ratio :

Options

A. `18:24:72`
B. `24:72:18`
C. `24:18:72`
D. `72:24:18`

Explanation

To solve the problem of finding the lengths of the segments of a stretched wire divided into three parts with frequencies in the ratio of 1:3:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Frequency and Length**:
The frequency of a vibrating string is inversely proportional to its length. This means that if the frequencies are in the ratio \( \nu_1 : \nu_2 : \nu_3 = 1 : 3 : 4 \), then the lengths will be in the inverse ratio:
\[ \frac{L_1}{L_2} = \frac{\nu_2}{\nu_1} = \frac{3}{1} \quad \text{and} \quad \frac{L_2}{L_3} = \frac{\nu_3}{\nu_2} = \frac{4}{3} \]

2. **Express Lengths in Terms of a Common Variable**:
Let the lengths of the segments be \( L_1, L_2, \) and \( L_3 \). From the ratios:
\[ L_2 = 3L_1 \quad \text{and} \quad L_3 = \frac{3}{4}L_2 = \frac{3}{4}(3L_1) = \frac{9}{4}L_1 \]

3. **Set Up the Equation for Total Length**:
The total length of the wire is given as 114 cm:
\[ L_1 + L_2 + L_3 = 114 \]
Substituting the expressions for \( L_2 \) and \( L_3 \):
\[ L_1 + 3L_1 + \frac{9}{4}L_1 = 114 \]

4. **Combine the Terms**:
Combine the terms on the left side:
\[ L_1 + 3L_1 + \frac{9}{4}L_1 = \left(1 + 3 + \frac{9}{4}\right)L_1 = \left(\frac{4}{4} + \frac{12}{4} + \frac{9}{4}\right)L_1 = \frac{25}{4}L_1 \]
Thus, we have:
\[ \frac{25}{4}L_1 = 114 \]

5. **Solve for \( L_1 \)**:
Multiply both sides by \( \frac{4}{25} \):
\[ L_1 = 114 \cdot \frac{4}{25} = \frac{456}{25} = 18.24 \text{ cm} \]

6. **Calculate \( L_2 \) and \( L_3 \)**:
Now, substituting \( L_1 \) back to find \( L_2 \) and \( L_3 \):
\[ L_2 = 3L_1 = 3 \cdot 18.24 = 54.72 \text{ cm} \]
\[ L_3 = \frac{9}{4}L_1 = \frac{9}{4} \cdot 18.24 = 40.56 \text{ cm} \]

7. **Find the Ratios**:
Now we can express the lengths in their simplest ratio form:
\[ L_1 : L_2 : L_3 = 18.24 : 54.72 : 40.56 \]
Dividing each length by \( 18.24 \):
\[ 1 : 3 : 2.22 \approx 1 : 3 : 2.2 \]

### Final Answer:
The lengths of the segments must be in the ratio \( 1 : 3 : 4 \).