1. Problem: The area of a circle is numerically equal to four times its circumference. Find the radius of the circle. 2. Formulae: - Area of a circle: $$A = \pi r^2$$ - Circumference of a circle: $$C = 2\pi r$$ 3. Given: $$A = 4C$$ 4. Substitute the formulas: $$\pi r^2 = 4 \times 2\pi r$$ 5. Simplify the right side: $$\pi r^2 = 8\pi r$$ 6. Divide both sides by $$\pi r$$ (assuming $$r \neq 0$$): $$\frac{\cancel{\pi} r^2}{\cancel{\pi} r} = \frac{8 \cancel{\pi} r}{\cancel{\pi} r}$$ $$r = 8$$ 7. Final answer: The radius of the circle is $$\boxed{8}$$ units. --- 2. Problem: Two arcs of the same circle subtend angles in the ratio 3:5. If the difference of their arc lengths is $$16\pi$$ cm, find the radius of the circle. 3. Formula for arc length: $$L = r \theta$$ where $$\theta$$ is in radians. 4. Let the angles be $$3x$$ and $$5x$$ radians. 5. Difference of arc lengths: $$r(5x) - r(3x) = 16\pi$$ $$r(5x - 3x) = 16\pi$$ $$2rx = 16\pi$$ 6. We need to find $$r$$ but we have two variables. Since angles are in ratio 3:5, total angle is $$3x + 5x = 8x$$. 7. The total angle around a circle is $$2\pi$$ radians, so: $$8x = 2\pi$$ $$x = \frac{2\pi}{8} = \frac{\pi}{4}$$ 8. Substitute $$x$$ back: $$2r \times \frac{\pi}{4} = 16\pi$$ 9. Simplify: $$\frac{2r\pi}{4} = 16\pi$$ $$\frac{r\pi}{2} = 16\pi$$ 10. Divide both sides by $$\pi$$: $$\frac{r}{2} = 16$$ 11. Multiply both sides by 2: $$r = 32$$ 12. Final answer: The radius of the circle is $$\boxed{32}$$ cm. --- 3. Problem: A sector has a central angle of 60º. If its area is equal to the area of a circle with radius 7 cm, find the radius of the sector. Use $$\pi=\frac{22}{7}$$. 4. Formula for sector area: $$A = \frac{\theta}{360} \pi r^2$$ where $$\theta$$ is in degrees. 5. Area of circle with radius 7: $$A_{circle} = \pi \times 7^2 = 49\pi$$ 6. Set sector area equal to circle area: $$\frac{60}{360} \pi r^2 = 49\pi$$ 7. Simplify fraction: $$\frac{1}{6} \pi r^2 = 49\pi$$ 8. Divide both sides by $$\pi$$: $$\frac{1}{6} r^2 = 49$$ 9. Multiply both sides by 6: $$r^2 = 294$$ 10. Take square root: $$r = \sqrt{294} = \sqrt{49 \times 6} = 7\sqrt{6}$$ 11. Final answer: The radius of the sector is $$\boxed{7\sqrt{6}}$$ cm. --- 4.1 Problem: Find the area of the shaded region consisting of 4 equal circles inside a square of side 6. 5. Each circle fits exactly in one quadrant of the square, so radius of each circle is half the side of the quadrant: $$r = \frac{6}{2} = 3$$ 6. Area of one circle: $$A = \pi r^2 = \pi \times 3^2 = 9\pi$$ 7. Total area of 4 circles: $$4 \times 9\pi = 36\pi$$ 8. Final answer: The total shaded area is $$\boxed{36\pi}$$. --- 4.2 Problem: Find the area of the shaded region inside a circle of diameter 30 that resembles a yin-yang shape formed by two symmetrical shapes. 9. The total area of the circle: $$r = \frac{30}{2} = 15$$ $$A = \pi r^2 = \pi \times 15^2 = 225\pi$$ 10. The yin-yang shape divides the circle into two equal shaded and unshaded parts. 11. Therefore, shaded area is half the circle area: $$\frac{1}{2} \times 225\pi = 112.5\pi$$ 12. Final answer: The shaded area is $$\boxed{112.5\pi}$$.