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

Options

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

Answer

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 first scenario
Given that \( x \) produces 8 beats per second with a frequency of 250 Hz, we can write:
\[ |x - 250| = 8 \]
This gives us two possible equations:
1. \( x - 250 = 8 \) (which simplifies to \( x = 258 \))
2. \( x - 250 = -8 \) (which simplifies to \( x = 242 \))

### Step 3: Set up the equations for the second scenario
Next, we know that \( x \) produces 12 beats per second with a frequency of 270 Hz, so we can write:
\[ |x - 270| = 12 \]
This gives us two possible equations:
1. \( x - 270 = 12 \) (which simplifies to \( x = 282 \))
2. \( x - 270 = -12 \) (which simplifies to \( x = 258 \))

### Step 4: Find the common frequency
From the first scenario, we found two possible values for \( x \): 258 Hz and 242 Hz. From the second scenario, we found two possible values for \( x \): 282 Hz and 258 Hz.

The common frequency between the two scenarios is:
\[ x = 258 \text{ Hz} \]

### Step 5: Conclusion
Thus, the unknown frequency \( x \) is 258 Hz.