1. The problem is to find the length of an arc for each given radius and central angle. 2. The formula for arc length is given by $$L = 2\pi r \cdot \left(\frac{\theta}{360}\right)$$ where $r$ is the radius and $\theta$ is the central angle in degrees. 3. Let's calculate each arc length step-by-step: **1)** $r=15$ m, $\theta=315^\circ$ $$L=2\pi \times 15 \times \frac{315}{360} = 30\pi \times 0.875 = 26.25\pi \approx 82.47\text{ m}$$ **2)** $r=12$ yd, $\theta=45^\circ$ $$L=2\pi \times 12 \times \frac{45}{360} = 24\pi \times 0.125 = 3\pi \approx 9.42\text{ yd}$$ **3)** $r=13$ mi, $\theta=285^\circ$ $$L=2\pi \times 13 \times \frac{285}{360} = 26\pi \times 0.7917 = 20.58\pi \approx 64.64\text{ mi}$$ **4)** $r=4$ ft, $\theta=195^\circ$ $$L=2\pi \times 4 \times \frac{195}{360} = 8\pi \times 0.5417 = 4.33\pi \approx 13.60\text{ ft}$$ **5)** $r=7$ mi, $\theta=120^\circ$ $$L=2\pi \times 7 \times \frac{120}{360} = 14\pi \times 0.3333 = 4.67\pi \approx 14.66\text{ mi}$$ **6)** $r=9$ ft, $\theta=315^\circ$ $$L=2\pi \times 9 \times \frac{315}{360} = 18\pi \times 0.875 = 15.75\pi \approx 49.48\text{ ft}$$ **7)** $r=18$ cm, $\theta=105^\circ$ $$L=2\pi \times 18 \times \frac{105}{360} = 36\pi \times 0.2917 = 10.5\pi \approx 32.99\text{ cm}$$ **8)** $r=16$ ft, $\theta=45^\circ$ $$L=2\pi \times 16 \times \frac{45}{360} = 32\pi \times 0.125 = 4\pi \approx 12.57\text{ ft}$$ Final answers: 1) $82.47$ m 2) $9.42$ yd 3) $64.64$ mi 4) $13.60$ ft 5) $14.66$ mi 6) $49.48$ ft 7) $32.99$ cm 8) $12.57$ ft