The angular diameter of a moon is 30 minute. At what distance a coin of diameter 2 . 2 cm should be placed so that moon will be fully covered.

Answer

To solve the problem, we need to determine the distance at which a coin with a diameter of 2.2 cm should be placed so that it fully covers the moon, given that the angular diameter of the moon is 30 minutes.

### Step-by-Step Solution:

1. **Convert Angular Diameter to Radians:**
The angular diameter of the moon is given as 30 minutes. We need to convert this to radians.
\[ \text{Angular diameter in degrees} = \frac{30}{60} = 0.5 \text{ degrees} \]
To convert degrees to radians, we use the formula:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
Therefore,
\[ \text{Angular diameter in radians} = 0.5 \times \frac{\pi}{180} = \frac{\pi}{360} \text{ radians} \]

2. **Use the Formula for Angular Size:**
The angular size (θ) of an object can be related to its diameter (D) and distance (d) from the observer using the formula:
\[ \theta = \frac{D}{d} \]
Rearranging this gives:
\[ d = \frac{D}{\theta} \]

3. **Substituting Values:**
Here, the diameter of the coin (D) is 2.2 cm, and the angular size (θ) is \(\frac{\pi}{360}\) radians.
\[ d = \frac{2.2}{\frac{\pi}{360}} = 2.2 \times \frac{360}{\pi} \]

4. **Calculating the Distance:**
Now we can calculate the value of \(d\):
\[ d \approx 2.2 \times \frac{360}{3.14} \approx 2.2 \times 114.59 \approx 252.12 \text{ cm} \]

5. **Final Answer:**
Therefore, the distance at which the coin should be placed to fully cover the moon is approximately:
\[ d \approx 252.12 \text{ cm} \]