1. **State the problem:** Two arcs of the same circle subtend angles in the ratio 3:5. The difference of their arc lengths is 16\pi cm. We need to find the radius of the circle. 2. **Recall the formula for arc length:** The length $L$ of an arc subtending an angle $\theta$ (in radians) in a circle of radius $r$ is given by: $$L = r\theta$$ 3. **Set up variables:** Let the angles subtended by the two arcs be $3x$ and $5x$ radians respectively. 4. **Express arc lengths:** - Arc length 1: $L_1 = r \times 3x = 3rx$ - Arc length 2: $L_2 = r \times 5x = 5rx$ 5. **Use the difference of arc lengths:** $$L_2 - L_1 = 16\pi$$ Substitute the expressions: $$5rx - 3rx = 16\pi$$ $$\cancel{r}x(5 - 3) = 16\pi$$ $$2rx = 16\pi$$ 6. **Simplify to find $rx$:** $$rx = \frac{16\pi}{2} = 8\pi$$ 7. **Use the fact that the total angle is $3x + 5x = 8x$ radians, which must be less than or equal to $2\pi$ for a circle:** Since the arcs are parts of the same circle, the total angle $8x$ should be $2\pi$ (full circle) or less. Assuming the arcs cover the full circle: $$8x = 2\pi$$ $$x = \frac{2\pi}{8} = \frac{\pi}{4}$$ 8. **Find the radius $r$ using $rx = 8\pi$:** $$r \times \frac{\pi}{4} = 8\pi$$ Divide both sides by $\pi$: $$r \times \frac{1}{4} = 8$$ $$\cancel{r} \times \frac{1}{4} = 8$$ $$r = 8 \times 4 = 32$$ **Final answer:** The radius of the circle is $\boxed{32}$ cm.