A Spaceship moves in a circular orbit of radius 7200 KM around the earth. How far does it travel while sweeping an angle of 〖100〗^0.

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.

Explore how to calculate the distance traveled by a spaceship in a circular orbit around Earth. Given a radius of 7200 km and an angle of 100 degrees, learn how to apply the arc length formula to find the distance traveled. Suitable for high school students studying geometry and trigonometry.

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