1. **Problem statement:** Find the radius of a cone when the curved surface area (CSA) is 286 cm² and the ratio of the radius to the slant height is 7:13. 2. **Formula for curved surface area of a cone:** $$\text{CSA} = \pi r l$$ where $r$ is the radius and $l$ is the slant height. 3. **Given ratio:** $$\frac{r}{l} = \frac{7}{13} \implies l = \frac{13}{7}r$$ 4. **Substitute $l$ in the CSA formula:** $$286 = \pi r \times \frac{13}{7}r = \pi \frac{13}{7} r^2$$ 5. **Solve for $r^2$:** $$r^2 = \frac{286 \times 7}{13 \pi} = \frac{2002}{13 \pi}$$ 6. **Calculate $r$:** $$r = \sqrt{\frac{2002}{13 \pi}}$$ 7. **Approximate value:** Using $\pi \approx 3.1416$, $$r = \sqrt{\frac{2002}{40.8408}} = \sqrt{49.01} \approx 7$$ **Final answer:** The radius of the cone is approximately **7 cm**.