The possible beat frequencies when A and C are sounded together are

Options

A. 2 Hz and 7 HZ
B. 1 Hz and 6 HZ
C. 1 Hz and 7 HZ
D. 2 Hz and 6 HZ

Answer

To find the possible beat frequencies when tuning forks A and C are sounded together, we can follow these steps:

### Step 1: Understand Beat Frequency
The beat frequency is the absolute difference between the frequencies of two sound sources. If two tuning forks have frequencies \( f_1 \) and \( f_2 \), the beat frequency \( f_b \) is given by:
\[ f_b = |f_1 - f_2| \]

### Step 2: Identify Given Frequencies
From the problem, we know:
- Fork B has a frequency of \( f_B = 440 \, \text{Hz} \).
- The beat frequency between A and B is \( 3 \, \text{Hz} \).
- The beat frequency between B and C is \( 4 \, \text{Hz} \).

### Step 3: Calculate Frequencies of A and C
Using the beat frequency information:
1. For A and B:
\[ f_A = f_B \pm 3 \, \text{Hz} \implies f_A = 440 \pm 3 \implies f_A = 437 \, \text{Hz} \text{ or } 443 \, \text{Hz} \]

2. For B and C:
\[ f_C = f_B \pm 4 \, \text{Hz} \implies f_C = 440 \pm 4 \implies f_C = 436 \, \text{Hz} \text{ or } 444 \, \text{Hz} \]

### Step 4: Find Possible Beat Frequencies for A and C
Now we need to calculate the beat frequencies for the combinations of \( f_A \) and \( f_C \):
- If \( f_A = 437 \, \text{Hz} \) and \( f_C = 436 \, \text{Hz} \):
\[ f_{AC} = |437 - 436| = 1 \, \text{Hz} \]

- If \( f_A = 437 \, \text{Hz} \) and \( f_C = 444 \, \text{Hz} \):
\[ f_{AC} = |437 - 444| = 7 \, \text{Hz} \]

- If \( f_A = 443 \, \text{Hz} \) and \( f_C = 436 \, \text{Hz} \):
\[ f_{AC} = |443 - 436| = 7 \, \text{Hz} \]

- If \( f_A = 443 \, \text{Hz} \) and \( f_C = 444 \, \text{Hz} \):
\[ f_{AC} = |443 - 444| = 1 \, \text{Hz} \]

### Step 5: List Possible Beat Frequencies
The possible beat frequencies when A and C are sounded together are:
- \( 1 \, \text{Hz} \)
- \( 7 \, \text{Hz} \)

### Conclusion
Thus, the possible beat frequencies when A and C are sounded together are \( 1 \, \text{Hz} \) and \( 7 \, \text{Hz} \).

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