Go-Cart One and Co-Cart Two are on a 360-foot circular track. There is a camera at the center of the track for each go-cart. Camera One is following Go-Cart One, and Camera Two is following Co-Cart Two. If the angle between the two cameras is 40 degrees, how far apart are the two go-carts?

To find the distance between the two go-carts, we can use the formula for the distance \( d \) between two points on a circle based on the central angle and the radius. The formula is: \[ d = 2r \sin(\frac{\theta}{2}) \] Here, the radius \( r \) of the track is \( 180 \) feet (since the circumference is \( 360 \) feet, \( r = \frac{360}{2\pi} \approx 57.3 \text{ feet} \)). The angle \( \theta \) is \( 40 \) degrees. Converting that to radians gives us approximately \( 0.698 \) radians. Substituting these values into the formula we get: \[ d = 2 \cdot 180 \sin(20) \] Calculating \( d \) yields: \[ d \approx 2 \cdot 180 \cdot 0.342 = 122.64 \text{ feet} \] So, the distance between Go-Cart One and Co-Cart Two is approximately **122.64 feet**. --- The world of go-carts isn’t just about racing; it’s rich with physics! Understanding real-world application of circular motion can explain why your favorite ride is so exhilarating! The concepts of radius, angles, and trigonometry help engineers design tracks that homeowners cannot resist! Ever wonder why your angle calculations sometimes seem off? A common pitfall in geometry and trigonometry is not converting degrees to radians when necessary! Misinterpreting these units can lead you astray, so always double-check! Stay sharp, and follow the track of math correctly!