Circular Track Arc Length Calculation
This problem requires calculating the length of a curved path (arc length) on a circular track given the track's radius and the angle the path subtends at the center.
Given Information
- Radius of the circular track, $r = 50 \text{ m}$.
- Angle subtended at the center, $\theta = 120^\circ$.
Formula for Arc Length
The length of an arc ($L$) in a circle is calculated using the formula:
$L = r \times \theta_{\text{radians}}$
where $r$ is the radius and $\theta_{\text{radians}}$ is the angle in radians.
Calculation Steps
- Convert Angle to Radians: The angle is given in degrees, but the formula requires radians. To convert degrees to radians, multiply by $\frac{\pi}{180^\circ}$. $ \theta_{\text{radians}} = 120^\circ \times \frac{\pi}{180^\circ} $ $ \theta_{\text{radians}} = \frac{120}{180} \pi = \frac{2}{3} \pi \text{ radians} $
- Calculate Arc Length: Now, substitute the radius and the angle in radians into the arc length formula. $ L = r \times \theta_{\text{radians}} $ $ L = 50 \text{ m} \times \frac{2\pi}{3} $ $ L = \frac{100\pi}{3} \text{ m} $
Result
The length of the arc the runner covers is $\frac{100\pi}{3}$ meters.
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