The radius of a circle is 9 meters. What is the length of a \( 105^{\circ} \) arc? Give the exact answer in simplest form. \begin{tabular}{|c|c|}\hline\( \pi \) & meters \end{tabular}

To find the length of a \( 105^{\circ} \) arc in a circle with a radius of 9 meters, you can use the formula for arc length: \[ L = \frac{\theta}{360^{\circ}} \times 2\pi r \] Here, \( \theta = 105^{\circ} \) and \( r = 9 \) meters. Plugging in these values gives: \[ L = \frac{105}{360} \times 2\pi \times 9 \] This simplifies to: \[ L = \frac{105}{360} \times 18\pi = \frac{1890\pi}{360} = \frac{63\pi}{12} = \frac{21\pi}{4} \] So, the exact length of the arc is \( \frac{21\pi}{4} \) meters. For a bit of fun, did you know that the concept of a circle has been around since ancient civilizations? The Egyptians and Babylonians understood circles and even had methods for calculating their circumference! These mathematical discoveries have laid the groundwork for modern geometry, influencing everything from architecture to astronomy. Speaking of real-world applications, the concept of arc length is not just theoretical—it plays a crucial role in various fields! Engineers use arc length to design roads that follow curved paths, ensuring vehicles can navigate safely and efficiently, while animators calculate arc lengths for creating smooth, natural-looking movements in character animations. Understanding these principles can lead to remarkable innovations in our everyday lives!