Samantha is riding a bike along a circular track. The track has a radius of 50 meters, and she rides along an arc that covers an angle of 240 degrees. How far does Samantha ride along the arc?

To calculate the distance Samantha rides along the arc, we use the formula for the length of an arc in a circle:

\[
L = r \theta
\]

Where:
- \(L\) is the arc length (the distance Samantha rides),
- \(r\) is the radius of the circle,
- \(\theta\) is the central angle of the arc in radians.

### Step 1: Convert the angle from degrees to radians
To convert 240 degrees to radians, we use the conversion factor:

\[
1^\circ = \frac{\pi}{180} \text{ radians}
\]

Thus,

\[
\theta = 240^\circ \times \frac{\pi}{180} = \frac{4\pi}{3} \text{ radians}
\]

### Step 2: Use the arc length formula
Now that we have \(\theta = \frac{4\pi}{3}\) radians and the radius \(r = 50\) meters, we can calculate the arc length:

\[
L = 50 \times \frac{4\pi}{3} = \frac{200\pi}{3}
\]

### Step 3: Approximate the value
Using \(\pi \approx 3.1416\), we get:

\[
L \approx \frac{200 \times 3.1416}{3} \approx 209.44 \text{ meters}
\]

### Conclusion:
Samantha rides approximately **209.44 meters** along the arc.

Would you like further clarification or additional details?

Here are some related questions to explore:
1. What is the total circumference of the track?
2. How long would Samantha ride if the angle were 360 degrees?
3. How would the arc length change if the radius were doubled?
4. What is the formula for arc length if the angle is in degrees instead of radians?
5. How could this concept apply to real-life scenarios, such as driving around a curved road?

**Tip:** Remember that radians and degrees are just different ways to measure angles, and they are interchangeable using the conversion \(\pi \text{ radians} = 180^\circ\).

Calculate the arc length covered by Samantha as she rides a bike along a circular track with a radius of 50 meters and an angle of 240 degrees. Learn key concepts such as converting degrees to radians and applying the arc length formula. Suitable for students in Grades 9-11.

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