A vibrating tuning fork is moving slowly and uniformly ins a horizontal circular path of radiu 8 m . The shortest distance of an observer ins ame plane from the tuning fork is 9m. The distance between the tuning fork and observer at the instant when apparent frequency becomes maximum is

Options

A. 9m
B. 25 m
C. 15 m
D. `sqrt(353)` m

Answer

To solve the problem, we need to determine the distance between the vibrating tuning fork and the observer at the instant when the apparent frequency is maximum. We will use the information given about the circular motion of the tuning fork and the position of the observer.

### Step-by-Step Solution:

1. **Understand the Setup**:
- The tuning fork is moving in a horizontal circular path with a radius of 8 meters.
- The shortest distance from the observer to the circular path of the tuning fork is 9 meters.

2. **Identify the Key Positions**:
- The tuning fork will be at various points along the circular path. The position where the apparent frequency is maximum occurs when the tuning fork is moving directly towards the observer.

3. **Visualize the Geometry**:
- The observer is located at a distance of 9 meters from the closest point on the circular path.
- The radius of the circular path is 8 meters.

4. **Calculate the Distance from the Observer to the Tuning Fork**:
- At the point where the tuning fork is closest to the observer, the distance can be calculated using the Pythagorean theorem.
- The distance from the observer to the center of the circular path (where the tuning fork is revolving) can be represented as:
\[ D = \sqrt{(radius)^2 + (shortest\ distance)^2} \]
- Here, the radius of the circular path is 8 meters, and the shortest distance to the observer is 9 meters.
- Thus, we have:
\[ D = \sqrt{8^2 + 9^2} = \sqrt{64 + 81} = \sqrt{145} \]

5. **Final Calculation**:
- The distance \(D\) from the observer to the tuning fork when the apparent frequency is maximum is:
\[ D = \sqrt{145} \text{ meters} \]

### Conclusion:
The distance between the tuning fork and the observer at the instant when the apparent frequency becomes maximum is \(\sqrt{145}\) meters.