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 $\pi \text{ radians} = 180^\circ$.