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The problem asks for calculations related to the orbit of planet XYZ around star ABC in a circular orbit with a radius of \( 82,000,000 \) miles. Let's break down each part:

**(a) Finding the angle formed by XYZ’s movement in one day (in radians):**

Given:
- The period of revolution of the planet XYZ around the star ABC is 378 days.
- One full revolution (the angle around the orbit) is \( 2\pi \) radians (since it's a circle).

To find the angle the planet moves in one day, we need to determine the fraction of the full revolution completed in one day.

The fraction is \( \frac{1 \text{ day}}{378 \text{ days}} \).

So, the angle in one day is:
\[
\theta = \frac{1}{378} \times 2\pi = \frac{2\pi}{378} = \frac{\pi}{189} \text{ radians}.
\]

Thus, the angle formed by XYZ’s movement in one day is \( \frac{\pi}{189} \) radians.

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Would you like to proceed with parts (b) and (c) next, or do you have any questions on part (a)?

Here are some related questions to expand your understanding:

1. How would the answer change if the orbit period was shorter or longer?
2. Can we convert the angle in radians to degrees for better intuition?
3. What would happen to the orbit angle if XYZ orbited in half the time?
4. Can this formula be applied to objects in elliptical orbits?
5. How can we express this angle per hour instead of per day?

**Tip**: In circular motion, angles in radians are proportional to the time taken compared to the orbital period.

The planet XYZ travels about the star ABC in an orbit that is almost circular. Assume that the orbit is a circle with a radius of 82,000,000 mi. Assume there are 24 hours in one day on planet XYZ. (a) Assume that XYZ planet year is 378 days, and find the angle formed by XYZ's movement in one day.

Learn how to calculate the angle formed by the movement of planet XYZ in one day as it orbits star ABC. Using the circular motion formula, solve for the angle in radians when given a radius of 82,000,000 miles and a period of 378 days. Suitable for high school students studying trigonometry and angular velocity.

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