Two swparate objects are situated from an eye at a distance of `10.8` km . The minimum distance between the two objects , for clear resolution, should be

Options

A. 10.8 m
B. 5.4 m
C. `2pim`
D. `pim`

Explanation

To solve the problem of finding the minimum distance between two objects for clear resolution by the human eye, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the minimum distance (D) between two objects that are 10.8 km away from the eye, such that they can be resolved clearly.

2. **Resolving Power of the Eye**:
The human eye can resolve objects that are separated by an angle of 1 arcminute (1/60th of a degree).

3. **Convert the Angle to Radians**:
To use the angle in calculations, we convert 1 arcminute to radians:
\[ \text{Angle in radians} = \frac{1}{60} \times \frac{\pi}{180} = \frac{\pi}{10800} \text{ radians} \]

4. **Using the Small Angle Approximation**:
The formula relating the angle (θ) in radians, the arc length (D), and the radius (r) is given by:
\[ \theta = \frac{D}{r} \]
Here, \( r \) is the distance from the eye to the objects (10.8 km).

5. **Convert Distance to Meters**:
Convert 10.8 km to meters:
\[ r = 10.8 \text{ km} = 10.8 \times 1000 = 10800 \text{ meters} \]

6. **Set Up the Equation**:
Substitute the values into the equation:
\[ \frac{\pi}{10800} = \frac{D}{10800} \]

7. **Solve for D**:
Rearranging gives:
\[ D = \frac{\pi}{10800} \times 10800 = \pi \text{ meters} \]

8. **Calculate the Value of D**:
Using the value of π (approximately 3.14):
\[ D \approx 3.14 \text{ meters} \]

### Final Answer:
The minimum distance between the two objects for clear resolution should be approximately **3.14 meters**.

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