The moon's distance from the earth is 360000 km and its diameter subtends an angle of 31' at the eye of the observer. Find the diameter of the moon.

Answer

To solve the problem of finding the diameter of the moon given its distance from the Earth and the angle it subtends, we can follow these steps:

### Step 1: Understand the Geometry
We have the moon at a distance of 360,000 km from the Earth, and it subtends an angle of 31 minutes at the observer's eye. We can visualize this as a triangle where the distance from the Earth to the moon is one side, and the diameter of the moon subtends the angle at the observer.

### Step 2: Convert the Angle from Minutes to Radians
First, we need to convert the angle from minutes to degrees, and then from degrees to radians.

- **Convert minutes to degrees:**
\[ \text{Angle in degrees} = \frac{31}{60} \text{ degrees} \]

- **Convert degrees to radians:**
\[ \text{Angle in radians} = \frac{31/60 \times \pi}{180} = \frac{31\pi}{10800} \text{ radians} \]

### Step 3: Use the Arc Length Formula
The arc length \( L \) can be calculated using the formula:
\[ L = r \theta \]
where:
- \( r \) is the radius (distance from the Earth to the moon, which is 360,000 km),
- \( \theta \) is the angle in radians.

Substituting the values:
\[ L = 360000 \times \frac{31\pi}{10800} \]

### Step 4: Calculate the Arc Length
Now we can compute the arc length:
\[ L = 360000 \times \frac{31\pi}{10800} \approx 3427.62 \text{ km} \]

### Step 5: Conclusion
The arc length \( L \) we calculated represents the diameter of the moon, which is approximately 3427.62 km.

### Final Answer
The diameter of the moon is approximately **3427.62 km**.

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