To determine how far Samantha rides along the arc, we need to calculate the arc length. The formula for the arc length $L$ of a circle is:

$L = r \theta$

where:

• $r$ is the radius of the circle,
• $\theta$ is the angle subtended by the arc at the center of the circle (in radians).

Step 1: Convert the angle from degrees to radians.

The angle is given as 240 degrees, but we need it in radians. The conversion from degrees to radians is done using the formula:

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

Substitute $\theta_{\text{degrees}} = 240$:

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

Step 2: Calculate the arc length.

Now that we have the angle in radians ($\theta = \frac{4\pi}{3}$) and the radius $r = 50$ meters, we can use the arc length formula:

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

Approximating $\pi$ as 3.1416:

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

Thus, Samantha rides approximately 209.44 meters along the arc.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the formula for the circumference of a circle?
2. How would the arc length change if the angle were 180 degrees?
3. Can you calculate the full circumference of the track Samantha is riding on?
4. How would the arc length change if the radius were doubled but the angle remained 240 degrees?
5. If Samantha completes one full lap (360 degrees), how far does she ride?

Tip: Always convert angles to radians when using trigonometric formulas unless otherwise specified.