The moon subtends an angle of 57 minutes at the base-line equal to the radius of the earth. What is the distance of the moon from the earth ? Radius of the earth = 6.4 x 10⁶ m.

The correct option is: 3.86x 10⁸ m.

Step-by-Step Explanation:

1. Understand the Given Information:
We are given:
- The angle subtended by the moon at the baseline of the earth, $\theta$ = 57 minutes (denoted as 57').
- The length of the baseline, which is equal to the radius of the earth, $b$ = $R$ = 6.4 x 10⁶ m.
We need to find the distance of the moon from the earth, $D$.

2. Convert the Angle into Radians:
To use the parallax formula, the angle $\theta$ must be in radians.
We know that:
- 1 degree = 60 minutes (60')
- ${180}^{\circ }$ = $\pi$ radians
So, 1 minute is:
$1\text{'}=\frac{1}{60}degree=\frac{1}{60}\times \frac{\pi }{180}radians$
Therefore, the angle $\theta$ in radians is:
$\theta =57\times \frac{\pi }{60\times 180}radians$
Using $\pi \approx 3.14159$:
$\theta \approx 57\times 0.000290888\approx 0.01658radians$

3. Apply the Distance Formula:
The relation between the baseline $b$, distance $D$, and subtended angle $\theta$ (in radians) is given by:
$\theta =\frac{b}{D}$
Rearranging this formula to solve for the distance $D$:
$D=\frac{b}{\theta }$

4. Calculate the Distance:
Substitute the values of baseline $b$ and angle $\theta$ into the equation:
$D=\frac{6.4\times {10}^6}{57\times \frac{3.14159}{60\times 180}}$
$D=\frac{6.4\times {10}^6\times 10800}{57\times 3.14159}$
$D=\frac{69120000}{179.07}\approx 385994\times {10}^{3}m$
Expressing it in scientific notation:
$D\approx 3.86\times {10}^{8}m$

Thus, the distance of the moon from the earth is approximately 3.86x 10⁸ m.