Two tuning forks, A and B, produce notes of frequencies 258 Hz and 262 Hz. An unknown note sounded with A produces certain beats. When the same note is sounded with B, the beat frequency gets doubled. The unknown frequency is

Options

A. 250 Hz
B. 252 Hz
C. 254 Hz
D. 256 Hz

Explanation

To solve the problem, we need to find the unknown frequency \( f \) based on the information provided about the tuning forks A and B.

### Step-by-Step Solution:

1. **Identify the frequencies of the tuning forks:**
- Frequency of tuning fork A, \( f_A = 258 \, \text{Hz} \)
- Frequency of tuning fork B, \( f_B = 262 \, \text{Hz} \)

2. **Define the unknown frequency:**
- Let the unknown frequency be \( f \).

3. **Calculate the beat frequencies:**
- The beat frequency when the unknown note is sounded with fork A is given by:
\[ \text{Beat frequency with A} = |f_A - f| = |258 - f| \]
- The beat frequency when the same note is sounded with fork B is given by:
\[ \text{Beat frequency with B} = |f_B - f| = |262 - f| \]

4. **Set up the relationship between the beat frequencies:**
- According to the problem, the beat frequency with fork B is double that with fork A:
\[ |262 - f| = 2 \times |258 - f| \]

5. **Consider the cases for the absolute values:**
- **Case 1:** Assume \( 262 - f = 2(258 - f) \)
\[ 262 - f = 516 - 2f \]
Rearranging gives:
\[ 2f - f = 516 - 262 \]
\[ f = 254 \, \text{Hz} \]

- **Case 2:** Assume \( 262 - f = -2(258 - f) \)
\[ 262 - f = -516 + 2f \]
Rearranging gives:
\[ 3f = 262 + 516 \]
\[ 3f = 778 \]
\[ f = \frac{778}{3} \approx 259.33 \, \text{Hz} \]

- **Case 3:** Assume \( f - 262 = 2(258 - f) \)
\[ f - 262 = 516 - 2f \]
Rearranging gives:
\[ 3f = 778 \]
\[ f = \frac{778}{3} \approx 259.33 \, \text{Hz} \]

- **Case 4:** Assume \( f - 262 = -2(258 - f) \)
\[ f - 262 = -516 + 2f \]
Rearranging gives:
\[ f = 256 \, \text{Hz} \]

6. **Evaluate the possible frequencies:**
- From the calculations, we have possible values for \( f \) as \( 254 \, \text{Hz} \) and \( 256 \, \text{Hz} \).

7. **Choose the correct answer from the options:**
- The options provided are:
- A) 250 Hz
- B) 252 Hz
- C) 254 Hz
- D) 256 Hz
- The correct answer based on our calculations is **C) 254 Hz**.