Assuming that the earth radius is 6400 km and that it subtends an angle of 57 at the center of the moon from the earth. find the distance between earth and moon center.

Answer

To find the distance between the Earth and the Moon's center given the Earth's radius and the angle subtended at the center of the Moon, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the Earth (R) = 6400 km
- Angle subtended at the center of the Moon (θ) = 57 minutes

2. **Convert the Angle from Minutes to Radians:**
- First, convert the angle from minutes to degrees:
\[ \text{Degrees} = \frac{57 \text{ minutes}}{60} = 0.95 \text{ degrees} \]
- Now convert degrees to radians using the formula:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
\[ \theta = 0.95 \times \frac{\pi}{180} \approx 0.0166 \text{ radians} \]

3. **Calculate the Distance (l) Using the Arc Length Formula:**
- The formula for arc length is:
\[ l = R \theta \]
- Here, R is the radius of the Earth and θ is the angle in radians:
\[ l = 6400 \times 0.0166 \approx 106.24 \text{ km} \]

4. **Final Calculation:**
- The distance between the Earth and the Moon's center is approximately 106.24 km.

### Final Answer:
The distance between the Earth and the Moon's center is approximately **106.24 km**.