1. Problem a) Calculate the area of the shaded sector with central angle $\alpha = 30^\circ$ in a circle of radius 4. 2. The area of a sector is given by the formula: $$\text{Area} = \pi r^2 \times \frac{\alpha}{360^\circ}$$ 3. Substitute $r=4$ and $\alpha=30^\circ$: $$\text{Area} = \pi \times 4^2 \times \frac{30}{360} = \pi \times 16 \times \frac{1}{12} = \frac{16\pi}{12} = \frac{4\pi}{3}$$ 4. Problem b) Calculate the area of the shaded region between two parallel chords $AB$ and $CD$ in a circle of radius 4, where $AB = 4\sqrt{3}$ and $CD = 4$. 5. The area between two chords parallel to each other can be found by subtracting the areas of the two circular segments formed by each chord. 6. First, find the distance from the center $O$ to each chord using the chord length formula: $$\text{Chord length} = 2\sqrt{r^2 - d^2}$$ where $d$ is the distance from the center to the chord. 7. For chord $AB$: $$4\sqrt{3} = 2\sqrt{16 - d_{AB}^2} \Rightarrow 2\sqrt{16 - d_{AB}^2} = 4\sqrt{3}$$ Divide both sides by 2: $$\sqrt{16 - d_{AB}^2} = 2\sqrt{3}$$ Square both sides: $$16 - d_{AB}^2 = 4 \times 3 = 12$$ $$d_{AB}^2 = 16 - 12 = 4 \Rightarrow d_{AB} = 2$$ 8. For chord $CD$: $$4 = 2\sqrt{16 - d_{CD}^2} \Rightarrow \sqrt{16 - d_{CD}^2} = 2$$ Square both sides: $$16 - d_{CD}^2 = 4 \Rightarrow d_{CD}^2 = 12 \Rightarrow d_{CD} = 2\sqrt{3}$$ 9. The area of a circular segment is: $$A = r^2 \arccos\left(\frac{d}{r}\right) - d \sqrt{r^2 - d^2}$$ 10. Calculate segment areas: - For $AB$: $$A_{AB} = 16 \arccos\left(\frac{2}{4}\right) - 2 \sqrt{16 - 4} = 16 \arccos\left(0.5\right) - 2 \times \sqrt{12}$$ $$= 16 \times \frac{\pi}{3} - 2 \times 2\sqrt{3} = \frac{16\pi}{3} - 4\sqrt{3}$$ - For $CD$: $$A_{CD} = 16 \arccos\left(\frac{2\sqrt{3}}{4}\right) - 2\sqrt{3} \sqrt{16 - 12} = 16 \arccos\left(\frac{\sqrt{3}}{2}\right) - 2\sqrt{3} \times 2$$ $$= 16 \times \frac{\pi}{6} - 4\sqrt{3} = \frac{16\pi}{6} - 4\sqrt{3} = \frac{8\pi}{3} - 4\sqrt{3}$$ 11. The shaded area between chords is: $$A = A_{AB} - A_{CD} = \left(\frac{16\pi}{3} - 4\sqrt{3}\right) - \left(\frac{8\pi}{3} - 4\sqrt{3}\right) = \frac{8\pi}{3}$$ 12. Problem c) Calculate the area of the shaded ring between two concentric circles with radii $r = 2\sqrt{2}$ and 4. 13. The area of the ring is the difference of the areas of the two circles: $$A = \pi R^2 - \pi r^2 = \pi (4^2) - \pi \left(2\sqrt{2}\right)^2 = 16\pi - \pi \times 8 = 8\pi$$ Final answers: - a) $\frac{4\pi}{3}$ - b) $\frac{8\pi}{3}$ - c) $8\pi$