The objective of telescope has diameter 12 cm. The distance at which two small green object placed 30 cm apart can be barely resolved by telescope, assuming the resolution to be limited by diffraction by objective only `(lamda=5.4xx10^(-5)cm)`

Options

A. 54.6 km
B. 57.0 km
C. 546 km
D. 52.4 km

Explanation

To solve the problem of determining the distance at which two small green objects can be barely resolved by a telescope, we will use the formula for angular resolution limited by diffraction. Here's a step-by-step breakdown:

### Step 1: Understand the Formula for Angular Resolution
The angular resolution (θ) can be expressed using the formula:
\[ \theta = \frac{1.22 \lambda}{D} \]
where:
- \( \lambda \) is the wavelength of light,
- \( D \) is the diameter of the telescope's objective.

### Step 2: Identify Given Values
From the problem, we have:
- Diameter of the telescope \( D = 12 \, \text{cm} = 0.12 \, \text{m} \)
- Wavelength \( \lambda = 5.4 \times 10^{-5} \, \text{cm} = 5.4 \times 10^{-7} \, \text{m} \)
- Distance between the two objects \( x = 30 \, \text{cm} = 0.3 \, \text{m} \)

### Step 3: Calculate the Angular Resolution
Using the formula for angular resolution:
\[ \theta = \frac{1.22 \times 5.4 \times 10^{-7}}{0.12} \]

### Step 4: Perform the Calculation
Calculating the numerator:
\[ 1.22 \times 5.4 \times 10^{-7} = 6.588 \times 10^{-7} \]
Now, substituting into the angular resolution formula:
\[ \theta = \frac{6.588 \times 10^{-7}}{0.12} = 5.49 \times 10^{-6} \, \text{radians} \]

### Step 5: Relate Angular Resolution to Distance
The distance \( D \) at which the two objects can be resolved is given by:
\[ D = \frac{x}{\theta} \]
Substituting the values:
\[ D = \frac{0.3}{5.49 \times 10^{-6}} \approx 54600 \, \text{m} = 54.6 \, \text{km} \]

### Final Answer
The distance at which the two small green objects can be barely resolved by the telescope is approximately:
\[ \boxed{54.6 \, \text{km}} \]