1. **Stating the problem:** We have a circular sector with radius $r=25$ meters and central angle $\theta=20^\circ$. The length $L=8.7$ meters and perimeter $P=58.7$ meters are given. 2. **Formula for arc length:** The arc length $L$ of a sector is given by $$L = r \times \theta_{\text{radians}}$$ where $\theta_{\text{radians}}$ is the central angle in radians. 3. **Convert angle to radians:** Since $1^\circ = \frac{\pi}{180}$ radians, $$\theta_{\text{radians}} = 20^\circ \times \frac{\pi}{180} = \frac{\pi}{9}$$ 4. **Calculate arc length:** $$L = 25 \times \frac{\pi}{9} = \frac{25\pi}{9} \approx 8.73 \text{ meters}$$ This matches the given $L=8.7$ meters, confirming the data. 5. **Formula for perimeter of sector:** The perimeter $P$ is the sum of the two radii and the arc length: $$P = 2r + L$$ 6. **Calculate perimeter:** $$P = 2 \times 25 + 8.7 = 50 + 8.7 = 58.7 \text{ meters}$$ This matches the given perimeter. **Final answer:** The arc length $L$ is approximately $8.7$ meters and the perimeter $P$ is $58.7$ meters, consistent with the formulas for a sector of radius 25 m and angle 20 degrees.