Number of beats between A and B is 5 and between B and C are 3. Number of beats between A and C may be

Options

A. 1
B. 2
C. 8
D. Both B and C

Explanation

To solve the problem of finding the number of beats between A and C, given the number of beats between A and B is 5 and between B and C is 3, we can follow these steps:

### Step 1: Understand the Concept of Beats
The number of beats is defined as the absolute difference in frequencies of two waves. If the number of beats between A and B is 5, we can express this mathematically as:
\[ |f_A - f_B| = 5 \]
This means that the frequency of A (f_A) can be either 5 units higher or lower than the frequency of B (f_B).

### Step 2: Set Up the Equations
From the information given:
1. \( f_A - f_B = 5 \) or \( f_B - f_A = 5 \)
2. The number of beats between B and C is 3, so:
\[ |f_B - f_C| = 3 \]
This gives us two more equations:
- \( f_B - f_C = 3 \) or \( f_C - f_B = 3 \)

### Step 3: Analyze the Cases
We will analyze the possible cases based on the relationships between the frequencies.

#### Case 1: \( f_A - f_B = 5 \) and \( f_B - f_C = 3 \)
From these equations:
- \( f_A = f_B + 5 \)
- \( f_B = f_C + 3 \)

Substituting the second equation into the first:
\[ f_A = (f_C + 3) + 5 = f_C + 8 \]
Thus, \( |f_A - f_C| = 8 \) (which means 8 beats).

#### Case 2: \( f_A - f_B = 5 \) and \( f_C - f_B = 3 \)
From these equations:
- \( f_A = f_B + 5 \)
- \( f_C = f_B + 3 \)

Substituting:
\[ f_A = (f_B - 3) + 5 = f_B + 2 \]
Thus, \( |f_A - f_C| = 2 \) (which means 2 beats).

#### Case 3: \( f_B - f_A = 5 \) and \( f_B - f_C = 3 \)
From these equations:
- \( f_B = f_A + 5 \)
- \( f_B = f_C + 3 \)

Substituting:
\[ f_C = f_B - 3 = (f_A + 5) - 3 = f_A + 2 \]
Thus, \( |f_A - f_C| = 2 \) (which means 2 beats).

#### Case 4: \( f_B - f_A = 5 \) and \( f_C - f_B = 3 \)
From these equations:
- \( f_B = f_A + 5 \)
- \( f_C = f_B + 3 \)

Substituting:
\[ f_C = (f_A + 5) + 3 = f_A + 8 \]
Thus, \( |f_A - f_C| = 8 \) (which means 8 beats).

### Step 4: Conclusion
From the analysis, we find that the number of beats between A and C can be either 2 or 8.

### Final Answer
The number of beats between A and C may be **2 or 8**.

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