A tuning fork of unknown frequency x, produces 5 beats per second with a tuning fork of frequency of 250 Hz. The produces 10 beats/second with another tuning fork frequency 265 Hz. The unknow frequency is

Options

A. 245 Hz
B. 255 Hz
C. 275 Hz
D. 270 Hz

Explanation

To find the unknown frequency \( x \) of the tuning fork, we can use the information provided about the beats produced with two different tuning forks. Here’s a step-by-step solution:

### Step 1: Understand the concept of beats
When two tuning forks of different frequencies are sounded together, they produce beats. The number of beats per second is equal to the absolute difference between their frequencies.

### Step 2: Set up the equations based on the given information
1. The first tuning fork has a frequency of 250 Hz and produces 5 beats per second with the unknown frequency \( x \). This gives us the equation:
\[ |x - 250| = 5 \]
This can lead to two possible equations:
\[ x - 250 = 5 \quad \text{(Equation 1)} \]
or
\[ 250 - x = 5 \quad \text{(Equation 2)} \]

2. The second tuning fork has a frequency of 265 Hz and produces 10 beats per second with the unknown frequency \( x \). This gives us the equation:
\[ |x - 265| = 10 \]
This can also lead to two possible equations:
\[ x - 265 = 10 \quad \text{(Equation 3)} \]
or
\[ 265 - x = 10 \quad \text{(Equation 4)} \]

### Step 3: Solve the equations
**From Equation 1:**
\[ x - 250 = 5 \implies x = 255 \quad \text{(Solution A)} \]

**From Equation 2:**
\[ 250 - x = 5 \implies x = 245 \quad \text{(Solution B)} \]

**From Equation 3:**
\[ x - 265 = 10 \implies x = 275 \quad \text{(Solution C)} \]

**From Equation 4:**
\[ 265 - x = 10 \implies x = 255 \quad \text{(Solution D)} \]

### Step 4: Analyze the solutions
Now, we have the possible values for \( x \):
- From Equation 1: \( x = 255 \)
- From Equation 2: \( x = 245 \)
- From Equation 3: \( x = 275 \)
- From Equation 4: \( x = 255 \)

### Step 5: Check for consistency
- If \( x = 255 \):
- For 250 Hz: \( |255 - 250| = 5 \) (correct)
- For 265 Hz: \( |255 - 265| = 10 \) (correct)

- If \( x = 245 \):
- For 250 Hz: \( |245 - 250| = 5 \) (correct)
- For 265 Hz: \( |245 - 265| = 20 \) (incorrect)

- If \( x = 275 \):
- For 250 Hz: \( |275 - 250| = 25 \) (incorrect)

### Conclusion
The only consistent solution is \( x = 255 \) Hz.

Thus, the unknown frequency is:
\[ \boxed{255 \text{ Hz}} \]