1. **Problem 10:** Calculate the perimeter of sector OAB with radius $r=7$ cm and area $40$ cm$^2$. 2. The area $A$ of a sector is given by $$A = \frac{1}{2} r^2 \theta$$ where $\theta$ is in radians. 3. Substitute $A=40$ and $r=7$: $$40 = \frac{1}{2} \times 7^2 \times \theta = \frac{49}{2} \theta$$ 4. Solve for $\theta$: $$\theta = \frac{40 \times 2}{49} = \frac{80}{49} \approx 1.6327 \text{ radians}$$ 5. The perimeter $P$ of the sector is the sum of two radii and the arc length: $$P = 2r + r\theta = 2 \times 7 + 7 \times 1.6327 = 14 + 11.429 = 25.429 \text{ cm}$$ 6. Round to 3 significant figures: $$P \approx 25.4 \text{ cm}$$ --- 7. **Problem 11:** Given a circle with center $O$; points $A$, $B$, and $C$ lie on the circle; length of arc $ABC = 5$ cm; angle $AOC = 55^{\circ}$. 8. Note: No question is explicitly asked. Assuming finding the radius $r$ of the circle. 9. Convert angle $AOC$ to radians: $$\theta = 55^{\circ} \times \frac{\pi}{180} = \frac{55\pi}{180} = \frac{11\pi}{36} \approx 0.9599$$ 10. The length of an arc is: $$s = r \theta$$ 11. Rearrange to solve for $r$: $$r = \frac{s}{\theta} = \frac{5}{0.9599} \approx 5.208 \text{ cm}$$ 12. Final answer: $$r \approx 5.21 \text{ cm}$$ (to 3 significant figures)