Viewed from Earth, the Moon’s angular size is about 0.52 degrees. Assume the distance
from the Earth to the Moon is 380, 000 kilometers. What is the diameter of the moon in miles?
(Hint: remember that units of size and distance must match.)

To solve this problem, we can use the formula for angular size, which relates the diameter of an object to its distance from the observer:

\[
\text{Angular size} = \frac{\text{Diameter of the object}}{\text{Distance to the object}}
\]

First, we'll convert the angular size from degrees to radians. The relationship between degrees and radians is given by:

\[
1 \text{ degree} = \frac{\pi}{180} \text{ radians}
\]

So, the angular size in radians is:

\[
\text{Angular size in radians} = 0.52 \times \frac{\pi}{180}
\]

Next, rearrange the formula for angular size to solve for the diameter of the Moon:

\[
\text{Diameter of the Moon} = \text{Angular size in radians} \times \text{Distance to the Moon}
\]

Finally, we'll convert the distance from kilometers to miles by using the conversion factor:

\[
1 \text{ kilometer} = 0.621371 \text{ miles}
\]

Let's calculate this step by step.The diameter of the Moon is approximately 2,143 miles.

Would you like more details or have any other questions? Here are some related questions for further exploration:

1. How does the angular size of the Moon change when it is closer to the Earth?
2. What is the formula for converting angular size from degrees to radians?
3. How does the distance to the Moon vary over time, and how does that affect its angular size?
4. How can you calculate the distance to an object if you know its angular size and actual diameter?
5. What is the significance of knowing the Moon's diameter in astronomy?

**Tip:** Understanding the relationship between angular size and distance is crucial for calculating the sizes of celestial objects seen from Earth.

Learn how to calculate the diameter of the Moon using its angular size and distance from Earth. This problem involves geometry, trigonometry, and unit conversion. Suitable for students in Grades 9-12.

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