1. **State the problem:** Calculate the perimeter of a sector of a circle with radius $3.5$ cm and central angle $35^\circ$. 2. **Formula for perimeter of a sector:** The perimeter $P$ of a sector is given by: $$P = 2r + l$$ where $r$ is the radius and $l$ is the length of the arc. 3. **Formula for arc length:** The arc length $l$ is given by: $$l = \frac{\theta}{360} \times 2\pi r$$ where $\theta$ is the central angle in degrees. 4. **Calculate the arc length:** $$l = \frac{35}{360} \times 2 \times \pi \times 3.5$$ $$= \frac{35}{360} \times 7\pi$$ 5. **Simplify the fraction:** $$\frac{35}{360} = \frac{\cancel{35}}{\cancel{360}} = \frac{7}{72}$$ 6. **Substitute back:** $$l = \frac{7}{72} \times 7\pi = \frac{49\pi}{72}$$ 7. **Calculate numerical value:** $$l \approx \frac{49 \times 3.1416}{72} \approx 2.14 \text{ cm}$$ 8. **Calculate perimeter:** $$P = 2r + l = 2 \times 3.5 + 2.14 = 7 + 2.14 = 9.14 \text{ cm}$$ **Final answer:** The perimeter of the sector is approximately **9.14 cm**.