An unknown frequency `x` produces 8 beats per seconds with a frequency of `250 Hz` and 12 beats with `270Hz` source. Then `x` is

Options

A. `258Hz`
B. `242Hz`
C. `262Hz`
D. `282Hz`

Explanation

To find the unknown frequency \( x \) that produces beats with the given frequencies, we can follow these steps:

### Step 1: Understand the concept of beats
Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two frequencies.

### Step 2: Set up the equations for the two scenarios
1. When the unknown frequency \( x \) produces 8 beats per second with a frequency of \( 250 \, \text{Hz} \):
\[ |x - 250| = 8 \]
This gives us two possible equations:
\[ x - 250 = 8 \quad \text{or} \quad 250 - x = 8 \]
Solving these, we get:
\[ x = 258 \quad \text{or} \quad x = 242 \]

2. When the unknown frequency \( x \) produces 12 beats per second with a frequency of \( 270 \, \text{Hz} \):
\[ |x - 270| = 12 \]
This gives us two possible equations:
\[ x - 270 = 12 \quad \text{or} \quad 270 - x = 12 \]
Solving these, we get:
\[ x = 282 \quad \text{or} \quad x = 258 \]

### Step 3: Find the common solution
Now we have two sets of possible values for \( x \):
- From the first scenario: \( x = 258 \) or \( x = 242 \)
- From the second scenario: \( x = 282 \) or \( x = 258 \)

The common frequency in both sets is:
\[ x = 258 \, \text{Hz} \]

### Step 4: Conclusion
Thus, the unknown frequency \( x \) is:
\[ \boxed{258 \, \text{Hz}} \]

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