A man wants to distinguish between two pillars located at a distance of `11 km`. What should be the minimum distance between the pillars ?

Explanation

To solve the problem of determining the minimum distance between two pillars that a man can distinguish when they are 11 km apart, we can follow these steps:

### Step 1: Understand the Concept of Resolution
The ability of the human eye to distinguish between two points is defined by the limit of resolution. For the human eye, this limit is approximately \( \theta = \frac{1}{60} \) degrees.

### Step 2: Convert the Angle to Radians
To work with this angle in calculations, we convert it from degrees to radians. The conversion is done using the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Thus, we have:
\[ \theta = \frac{1}{60} \times \frac{\pi}{180} = \frac{\pi}{10800} \text{ radians} \]

### Step 3: Set Up the Relationship Between Distance and Angle
The relationship between the minimum distance \( x \) between the two pillars, the distance \( d \) from the observer to the pillars, and the angle \( \theta \) is given by:
\[ \theta \approx \frac{x}{d} \]
where \( d \) is the distance to the pillars (11 km in this case).

### Step 4: Convert Distance to Meters
Convert the distance \( d \) from kilometers to meters:
\[ d = 11 \text{ km} = 11 \times 1000 = 11000 \text{ meters} \]

### Step 5: Solve for Minimum Distance \( x \)
Rearranging the equation \( \theta \approx \frac{x}{d} \) gives us:
\[ x = d \times \theta \]
Substituting the values we have:
\[ x = 11000 \times \frac{\pi}{10800} \]

### Step 6: Calculate the Value of \( x \)
Now we can calculate \( x \):
\[ x = 11000 \times \frac{\pi}{10800} \approx 11000 \times 0.000290888 \approx 3.2 \text{ meters} \]

### Conclusion
Thus, the minimum distance between the two pillars that the man can distinguish is approximately \( 3.2 \text{ meters} \).

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