Let's go through each part of the problem step by step.

Given:

• Orbit radius: $r = 90,000,000$ miles.
• Planet XYZ's year: 380 days.
• One day on planet XYZ: 24 hours.

Part (a) — The angle formed by XYZ's movement in one day

The planet XYZ completes a full revolution around its star ABC in one year. The total angle for a full revolution is $2\pi$ radians. Since the planet takes 380 days to complete one revolution, we can find the angle it moves in one day by dividing $2\pi$ by the number of days in the year.

The angle per day is:

$\text{Angle per day} = \frac{2\pi}{380}$

Now, the question asks for the angle in terms of radians, so this simplifies to:

$\text{Angle per day} = \frac{\pi}{190} \text{ radians}$

So, the angle formed by XYZ’s movement in one day is:

$\boxed{\frac{\pi}{190}} \text{ radians}$

Part (b) — The angular speed in radians per hour

We know the planet moves through $\frac{\pi}{190}$ radians per day. To find the angular speed in radians per hour, we divide the angle per day by the number of hours in one day:

$\text{Angular speed per hour} = \frac{\frac{\pi}{190}}{24}$

Simplifying:

$\text{Angular speed per hour} = \frac{\pi}{4560} \text{ radians per hour}$

So, the angular speed is:

$\boxed{\frac{\pi}{4560}} \text{ radians per hour}$

Part (c) — The linear speed in miles per hour

The linear speed $v$ of an object in circular motion is given by the formula:

$v = r \cdot \omega$

Where:

• $r$ is the radius of the orbit ($90,000,000$ miles),
• $\omega$ is the angular speed in radians per hour.

We already found that the angular speed is $\frac{\pi}{4560}$ radians per hour.

So the linear speed is:

$v = 90,000,000 \cdot \frac{\pi}{4560}$

Now, we calculate this:

$v \approx 90,000,000 \cdot \frac{3.1416}{4560} \approx 90,000,000 \cdot 0.000690 \approx 62,100 \text{ miles per hour}$

Thus, the linear speed of XYZ is approximately:

$\boxed{62,100} \text{ miles per hour}$

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Recap of the answers:

1. The angle formed by XYZ's movement in one day is $\frac{\pi}{190}$ radians.
2. The angular speed is $\frac{\pi}{4560}$ radians per hour.
3. The linear speed is approximately $62,100$ miles per hour.

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Would you like further explanation or have additional questions?

Here are some related questions for further practice:

1. How would the angle formed by XYZ's movement change if the year was 400 days instead of 380 days?
2. If the radius of XYZ's orbit were doubled, how would that affect the linear speed?
3. What would the linear speed be if the number of days in a year were halved to 190 days?
4. How can you calculate the period of revolution in days if you were given the angular speed in radians per day instead of radians per hour?
5. What is the relationship between the radius of an orbit and the linear speed in circular motion?

Tip: For problems involving circular motion, remember that the angular speed and linear speed are related by the formula $v = r \cdot \omega$, where $\omega$ is the angular speed in radians per unit of time, and $r$ is the radius of the orbit.