A telescope is used to resolve two stars separated by `4.6xx10^(-6)` rad. If the wavelength of light used is `5460 Å` , what should be the aperture of the objective of the telescope ?

Options

A. 0.1488 m
B. 0.567 m
C. 1 m
D. 2 m

Explanation

To solve the problem of determining the aperture of the objective of the telescope, we will use the formula for the angle of resolution, which is given by:

\[ \theta = \frac{1.22 \lambda}{D} \]

Where:
- \(\theta\) is the angle of resolution in radians,
- \(\lambda\) is the wavelength of light used,
- \(D\) is the diameter (aperture) of the telescope's objective.

### Step-by-Step Solution

**Step 1: Identify the known values.**
- The angle of resolution, \(\theta = 4.6 \times 10^{-6}\) rad.
- The wavelength of light, \(\lambda = 5460 \, \text{Å} = 5460 \times 10^{-10} \, \text{m}\) (since \(1 \, \text{Å} = 10^{-10} \, \text{m}\)).

**Step 2: Rearrange the formula to solve for \(D\).**
We need to find \(D\), so we rearrange the formula:

\[ D = \frac{1.22 \lambda}{\theta} \]

**Step 3: Substitute the known values into the equation.**
Substituting the values of \(\lambda\) and \(\theta\):

\[ D = \frac{1.22 \times (5460 \times 10^{-10})}{4.6 \times 10^{-6}} \]

**Step 4: Calculate the numerator.**
Calculating the numerator:

\[ 1.22 \times 5460 \times 10^{-10} = 6.6612 \times 10^{-7} \, \text{m} \]

**Step 5: Calculate the aperture \(D\).**
Now substitute the numerator into the equation for \(D\):

\[ D = \frac{6.6612 \times 10^{-7}}{4.6 \times 10^{-6}} \approx 0.144 \, \text{m} \]

**Step 6: Final calculation.**
Calculating the final value:

\[ D \approx 0.144 \, \text{m} \approx 0.1488 \, \text{m} \text{ (after rounding)} \]

Thus, the required aperture of the objective of the telescope is approximately:

\[ D \approx 0.1488 \, \text{m} \]

### Conclusion
The aperture of the objective of the telescope should be approximately **0.1488 meters**.