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`

Answer

To solve the problem, we need to find the lengths of the segments of a stretched wire based on the given frequency ratios. The steps are as follows:

### Step 1: Understand the relationship between frequency and length
The fundamental frequency of a stretched wire is inversely proportional to its length. This means that if the frequency increases, the length decreases, and vice versa.

### Step 2: Set up the frequency ratio
We are given the frequency ratio of the three segments as:
\[ F_1 : F_2 : F_3 = 1 : 3 : 4 \]

### Step 3: Write the inverse relationship for lengths
Since the frequency is inversely proportional to length, we can write:
\[ L_1 : L_2 : L_3 = \frac{1}{F_1} : \frac{1}{F_2} : \frac{1}{F_3} \]

### Step 4: Substitute the frequency values
Substituting the frequency values into the length ratio gives:
\[ L_1 : L_2 : L_3 = \frac{1}{1} : \frac{1}{3} : \frac{1}{4} \]

### Step 5: Simplify the ratios
To simplify the ratios, we can find a common denominator. The least common multiple (LCM) of the denominators (1, 3, and 4) is 12. Thus, we can express the lengths as:
\[ L_1 : L_2 : L_3 = 12 : 4 : 3 \]

### Step 6: Finalize the ratio
To express the ratio in a more standard form, we can divide each term by the smallest value, which is 3:
\[ L_1 : L_2 : L_3 = 4 : 1.33 : 1 \]

However, since we want the lengths in whole numbers, we can multiply through by 3 to avoid fractions:
\[ L_1 : L_2 : L_3 = 12 : 4 : 3 \]

### Step 7: Check against options
Upon checking the options provided in the question, we find that the ratio can be simplified to match one of the options, confirming that:
\[ L_1 : L_2 : L_3 = 72 : 24 : 18 \]

### Conclusion
Thus, the lengths of the segments must be in the ratio:
\[ 12 : 4 : 3 \]