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We are given the following details:
- The planet XYZ orbits star ABC in a nearly circular orbit with a radius of \( 82,000,000 \) miles.
- A day has 24 hours.
- The planet XYZ takes 378 days to complete one revolution (one full orbit) around the star.

We are tasked with finding:

### (a) The angle formed by XYZ’s movement in one day
To calculate the angle formed in one day, we need to understand that XYZ completes one full revolution (360 degrees or \( 2\pi \) radians) in 378 days. Therefore, the angle moved in one day is:

\[
\theta = \frac{2\pi}{378} \text{ radians/day}
\]

### (b) The angular speed in radians per hour
To find the angular speed per hour, we take the angular speed per day (from part a) and divide it by 24 (since there are 24 hours in a day):

\[
\omega = \frac{\theta}{24} = \frac{2\pi}{378 \times 24} \text{ radians/hour}
\]

### (c) The linear speed in miles per hour
The linear speed \( v \) can be calculated using the formula:

\[
v = r \cdot \omega
\]

Where:
- \( r = 82,000,000 \) miles (the radius of the orbit),
- \( \omega \) is the angular speed in radians per hour (from part b).

Let me calculate the exact values for each part now.Here are the results for each part:

### (a) The angle formed by XYZ's movement in one day:
\[
\theta \approx 0.0166 \text{ radians/day}
\]

### (b) The angular speed in radians per hour:
\[
\omega \approx 0.000693 \text{ radians/hour}
\]

### (c) The linear speed of XYZ in miles per hour:
\[
v \approx 56,792 \text{ miles/hour}
\]

Would you like further clarification on any of these steps or more details? Here are some additional related questions:

1. How would the angular speed change if the orbit radius was doubled?
2. How is the linear speed related to the radius of the orbit?
3. What would happen to the angular speed if the planet completed the orbit in fewer days?
4. How can we convert angular speed from radians per hour to degrees per hour?
5. What would be the linear speed if the planet's orbital radius was halved?

**Tip:** To easily convert angular speed from radians to degrees, remember that \( 2\pi \) radians is equivalent to 360 degrees.

The planet XYZ travels about the star ABC in an orbit that is almost circular. Assume that the orbit is a circle with radius 82,000,000 miles. 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. (b) Give the angular speed in radians per hour. (c) Find the linear speed of XYZ in miles per hour.

Learn how to calculate the angle formed by a planet's movement, its angular speed in radians per hour, and linear speed in miles per hour. Given a radius of 82,000,000 miles and a planetary year of 378 days, this problem uses formulas for angular velocity and linear speed.

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