To find out how far the spaceship travels while sweeping an angle of $100^\circ$ in a circular orbit of radius 7200 km, we can use the formula for the arc length of a circle. The formula is:

$\text{Arc length} = r \theta$

where $r$ is the radius of the circle and $\theta$ is the angle in radians.

First, we need to convert the angle from degrees to radians. We use the conversion factor $\pi \, \text{radians} = 180^\circ$:

$\theta = 100^\circ \times \frac{\pi}{180^\circ} = \frac{100\pi}{180} = \frac{5\pi}{9} \, \text{radians}$

Now we can calculate the arc length:

$\text{Arc length} = 7200 \, \text{km} \times \frac{5\pi}{9}$

$\text{Arc length} = 7200 \times \frac{5\pi}{9}$

$\text{Arc length} = 7200 \times \frac{5 \times 3.14159}{9}$

$\text{Arc length} = 7200 \times 1.74533$

$\text{Arc length} \approx 12566.4 \, \text{km}$

Therefore, the spaceship travels approximately 12566.4 km while sweeping an angle of $100^\circ$.

Would you like further details or have any questions?

Here are some more questions you might consider:

1. How do you convert degrees to radians?
2. What is the formula for the circumference of a circle?
3. How do you find the length of an arc if the angle is given in radians?
4. Can you explain the concept of a radian?
5. How would the arc length change if the radius of the orbit increased?

Tip: When dealing with angles in circular motion, always ensure you convert degrees to radians if the formula requires radians.