The moon's distance from the earth is `360000 km` and its diameter substends an angle of `42` at the eye of the observer. The diameter of the moon in kilometers is

Options

A. `4400`
B. `1000`
C. `3600`
D. `8800`

Explanation

To find the diameter of the moon based on the given information, we can follow these steps:

### Step 1: Convert the angle from minutes to degrees
The angle given is 42 minutes. Since 1 degree = 60 minutes, we can convert the angle in minutes to degrees:
\[ \text{Angle in degrees} = \frac{42}{60} = 0.7 \text{ degrees} \]

### Step 2: Convert the angle from degrees to radians
To use the formula relating the arc length, radius, and angle, we need to convert the angle from degrees to radians. The conversion formula is:
\[ \text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180} \]
So,
\[ \text{Angle in radians} = 0.7 \times \frac{\pi}{180} \]

### Step 3: Use the formula for arc length
The relationship between the arc length (which is the diameter of the moon, \(d\)), the radius (\(r\)), and the angle in radians (\(\theta\)) is given by:
\[ \theta = \frac{d}{r} \]
Rearranging this gives:
\[ d = r \times \theta \]
Here, \(r\) is the distance from the Earth to the Moon, which is given as \(360,000 \text{ km}\).

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ d = 360,000 \times \left(0.7 \times \frac{\pi}{180}\right) \]

### Step 5: Calculate the diameter
Now we can calculate \(d\):
1. First, calculate \(\theta\):
\[ \theta = 0.7 \times \frac{22}{7 \times 180} \approx 0.7 \times 0.000385 = 0.0002695 \text{ radians} \]
2. Now calculate \(d\):
\[ d = 360,000 \times 0.0002695 \approx 97.42 \text{ km} \]

### Final Result
The diameter of the moon is approximately \(97.42 \text{ km}\).