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

$L = r \theta$

Where:

• $L$ is the arc length.
• $r$ is the radius of the circle.
• $\theta$ is the central angle in radians.

Step 1: Convert the angle from degrees to radians

Since the given angle is in degrees, we need to convert it to radians. The conversion formula is:

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$

For $\theta = 240^\circ$:

$\theta_{\text{radians}} = 240 \times \frac{\pi}{180} = \frac{4\pi}{3} \, \text{radians}$

Step 2: Use the arc length formula

Now that we have $\theta$ in radians, we can substitute into the arc length formula:

$L = 50 \times \frac{4\pi}{3}$

Step 3: Simplify

$L = \frac{200\pi}{3} \, \text{meters}$

Using $\pi \approx 3.1416$:

$L \approx \frac{200 \times 3.1416}{3} \approx 209.44 \, \text{meters}$

Conclusion:

Samantha rides approximately 209.44 meters along the arc.

Would you like more details on this or have any questions?

Here are 5 related questions for further practice:

1. If the radius of the track was 30 meters instead, how far would Samantha have ridden?
2. How would the arc length change if Samantha covered an angle of 180 degrees?
3. What would be the distance if Samantha rode along the entire circular track?
4. How would you find the area of the sector formed by the arc and the radius of the circle?
5. If Samantha increased her speed, how long would it take her to complete the arc at 5 m/s?

Tip: Always convert angles into radians when dealing with arc lengths in circular geometry.