There is a set of four tuning forks , one with the lowest frequency vibrating at `550 Hz`. By using any two tuning forks at a time , the following beat frequencies are heard : `1 , 2, 3, 5 , 7 , 8`. The possible frequencies of the other three forks are

Options

A. `552 , 553 , 560`
B. `557, 558 , 560`
C. `552 , 553 ,558`
D. `551 , 553 , 558`

Explanation

To solve the problem of determining the possible frequencies of the other three tuning forks, we can follow these steps:

### Step 1: Identify the known frequency
We know that the lowest frequency tuning fork (let's call it \( f_1 \)) has a frequency of \( 550 \, \text{Hz} \).

### Step 2: Understand beat frequency
The beat frequency is defined as the absolute difference between the frequencies of two tuning forks. If we have two frequencies \( f_a \) and \( f_b \) (where \( f_a > f_b \)), the beat frequency \( f_{beat} \) is given by:
\[ f_{beat} = |f_a - f_b| \]

### Step 3: List the given beat frequencies
The beat frequencies we need to achieve are \( 1 \, \text{Hz}, 2 \, \text{Hz}, 3 \, \text{Hz}, 5 \, \text{Hz}, 7 \, \text{Hz}, \) and \( 8 \, \text{Hz} \).

### Step 4: Define the unknown frequencies
Let the frequencies of the other three tuning forks be \( f_2, f_3, \) and \( f_4 \).

### Step 5: Set up equations based on beat frequencies
Using \( f_1 = 550 \, \text{Hz} \), we can derive equations for the beat frequencies involving \( f_1 \):
- \( |f_2 - 550| \) must equal one of the beat frequencies
- \( |f_3 - 550| \) must equal one of the beat frequencies
- \( |f_4 - 550| \) must equal one of the beat frequencies

### Step 6: Determine possible values for \( f_2, f_3, \) and \( f_4 \)
1. **For \( f_2 \)**:
- If \( f_2 - 550 = 1 \) then \( f_2 = 551 \)
- If \( 550 - f_2 = 1 \) then \( f_2 = 549 \)
- If \( f_2 - 550 = 2 \) then \( f_2 = 552 \)
- If \( 550 - f_2 = 2 \) then \( f_2 = 548 \)
- If \( f_2 - 550 = 3 \) then \( f_2 = 553 \)
- If \( 550 - f_2 = 3 \) then \( f_2 = 547 \)
- If \( f_2 - 550 = 5 \) then \( f_2 = 555 \)
- If \( 550 - f_2 = 5 \) then \( f_2 = 545 \)
- If \( f_2 - 550 = 7 \) then \( f_2 = 557 \)
- If \( 550 - f_2 = 7 \) then \( f_2 = 543 \)
- If \( f_2 - 550 = 8 \) then \( f_2 = 558 \)
- If \( 550 - f_2 = 8 \) then \( f_2 = 542 \)

2. **For \( f_3 \)** and **\( f_4 \)**, we follow the same logic.

### Step 7: Check combinations to match all beat frequencies
We need to ensure that we can find three frequencies \( f_2, f_3, f_4 \) such that all the required beat frequencies can be produced.

### Step 8: Evaluate the options
Given the options:
- **Option A**: \( 552, 553, 560 \)
- **Option B**: \( 557, 558, 560 \)
- **Option C**: \( 552, 553, 558 \)
- **Option D**: \( 551, 553, 558 \)

We will check which option allows us to achieve all the required beat frequencies.

### Step 9: Verify Option D
1. \( f_2 = 551 \)
- \( |551 - 550| = 1 \)
2. \( f_3 = 553 \)
- \( |553 - 550| = 3 \)
3. \( f_4 = 558 \)
- \( |558 - 550| = 8 \)

Now check combinations:
- \( |553 - 551| = 2 \)
- \( |558 - 551| = 7 \)
- \( |558 - 553| = 5 \)

All required beat frequencies \( 1, 2, 3, 5, 7, 8 \) are accounted for.

### Conclusion
The possible frequencies of the other three tuning forks are:
\[ \text{Option D: } 551 \, \text{Hz}, 553 \, \text{Hz}, 558 \, \text{Hz} \]