1. The problem states that the volume of a cone is 565.7 and the height is 15 cm. We need to find the radius and the surface area of the cone. 2. The volume $V$ of a cone is given by the formula $$V = \frac{1}{3} \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. Plug in the known values: $$565.7 = \frac{1}{3} \pi r^2 (15)$$ 4. Simplify the equation: $$565.7 = 5 \pi r^2$$ 5. Solve for $r^2$: $$r^2 = \frac{565.7}{5 \pi}$$ 6. Calculate $r^2$: $$r^2 \approx \frac{565.7}{15.70796} \approx 36.03$$ 7. Find radius $r$: $$r = \sqrt{36.03} \approx 6$$ cm 8. Next, calculate the slant height $l$ using the Pythagorean theorem: $$l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 15^2} = \sqrt{36 + 225} = \sqrt{261} \approx 16.16$$ cm 9. The surface area $A$ of a cone is given by: $$A = \pi r^2 + \pi r l = \pi r (r + l)$$ 10. Substitute the values: $$A \approx \pi \times 6 (6 + 16.16) = \pi \times 6 \times 22.16 \approx 417.06$$ cm$^2$ Final answers: - Radius $r \approx 6$ cm - Surface area $A \approx 417.06$ cm$^2$