A person is viewing two red dots at a distance of `1000 m`. What should be the minimum distance between dots so the his eye can resolce two dots separately.
`("take" : lambda_(red) + 700 nm, "pupil dia" = 2.5 mm)`

Options

A. `45 cm`
B. `50 cm`
C. `35 cm`
D. `10 cm`

Explanation

To solve the problem of determining the minimum distance between two red dots so that a person can resolve them separately, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find the minimum distance \( D_0 \) between two dots that are viewed at a distance \( L = 1000 \, \text{m} \).

2. **Use the Angular Resolution Formula**:
The minimum angle \( \theta \) that the eye can resolve is given by the formula:
\[ \theta_{\text{min}} = \frac{1.22 \lambda}{D} \]
where \( \lambda \) is the wavelength of light and \( D \) is the diameter of the pupil.

3. **Convert Units**:
Given that \( \lambda = 700 \, \text{nm} = 700 \times 10^{-9} \, \text{m} \) and \( D = 2.5 \, \text{mm} = 2.5 \times 10^{-3} \, \text{m} \).

4. **Set Up the Relationship**:
The relationship between the distance \( D_0 \) between the dots and the angle \( \theta \) can be expressed as:
\[ D_0 = L \cdot \theta \]
Substituting the expression for \( \theta_{\text{min}} \):
\[ D_0 = L \cdot \frac{1.22 \lambda}{D} \]

5. **Substitute Known Values**:
Plugging in the values:
\[ D_0 = 1000 \cdot \frac{1.22 \times 700 \times 10^{-9}}{2.5 \times 10^{-3}} \]

6. **Calculate the Result**:
First, calculate \( 1.22 \times 700 \times 10^{-9} \):
\[ 1.22 \times 700 = 854 \times 10^{-9} \, \text{m} \]
Now, substitute this back into the equation:
\[ D_0 = 1000 \cdot \frac{854 \times 10^{-9}}{2.5 \times 10^{-3}} = 1000 \cdot \frac{854 \times 10^{-9}}{2.5 \times 10^{-3}} \]
Simplifying:
\[ D_0 = 1000 \cdot \frac{854}{2.5} \times 10^{-6} = 1000 \cdot 341.6 \times 10^{-6} \]
\[ D_0 = 0.3416 \, \text{m} = 34.16 \, \text{cm} \]

7. **Final Result**:
The minimum distance \( D_0 \) between the two red dots should be approximately \( 0.35 \, \text{m} \) or \( 35 \, \text{cm} \).