A telescope of aperture diameter 5m is used to observe the moon from the earth. Distance between the moon and earth is `4 xx 10^5` km. Determine the minimum distance between two points on the moon's surface which can be resolved using this telescope. (Wave length of light is `5893 A^@`.

Options

A. 200 m
B. 20m
C. 60 m
D. 600 m

Answer

To solve the problem of determining the minimum distance between two points on the moon's surface that can be resolved using a telescope with a given aperture diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Aperture diameter of the telescope, \( a = 5 \, \text{m} \)
- Distance from Earth to the Moon, \( r = 4 \times 10^5 \, \text{km} = 4 \times 10^8 \, \text{m} \) (convert kilometers to meters)
- Wavelength of light, \( \lambda = 5893 \, \text{Å} = 5893 \times 10^{-10} \, \text{m} \) (convert angstroms to meters)

2. **Use the Rayleigh Criterion:**
The minimum resolvable angle \( \theta \) (in radians) for a telescope is given by the Rayleigh criterion:
\[ \theta = \frac{1.22 \lambda}{a} \]

3. **Calculate the Minimum Resolving Angle:**
Substitute the values of \( \lambda \) and \( a \):
\[ \theta = \frac{1.22 \times 5893 \times 10^{-10}}{5} \]

4. **Perform the Calculation:**
- Calculate \( 1.22 \times 5893 \times 10^{-10} \):
\[ 1.22 \times 5893 \approx 7192.56 \times 10^{-10} \, \text{m} \]
- Now divide by the aperture diameter:
\[ \theta \approx \frac{7192.56 \times 10^{-10}}{5} \approx 1438.512 \times 10^{-10} \, \text{radians} \]

5. **Calculate the Minimum Distance on the Moon's Surface:**
The minimum distance \( d \) between two points on the moon's surface that can be resolved is given by:
\[ d = r \cdot \theta \]
Substitute \( r \) and \( \theta \):
\[ d = 4 \times 10^8 \cdot 1438.512 \times 10^{-10} \]

6. **Perform the Final Calculation:**
\[ d \approx 4 \times 10^8 \cdot 1.438512 \times 10^{-7} \approx 57.54 \, \text{m} \]

7. **Round to the Nearest Whole Number:**
The minimum distance that can be resolved is approximately \( 57.54 \, \text{m} \), which can be rounded to \( 60 \, \text{m} \).

### Final Answer:
The minimum distance between two points on the moon's surface that can be resolved using this telescope is approximately **60 meters**.