1. **State the problem:** We need to find the volume of a cylinder with a radius of 8.4 cm and a radius-to-height ratio of 2:6. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Use the ratio to find the height:** Given the ratio $\frac{r}{h} = \frac{2}{6} = \frac{1}{3}$, so $$h = 3r$$ 4. **Calculate the height:** $$h = 3 \times 8.4 = 25.2 \text{ cm}$$ 5. **Calculate the volume:** $$V = \pi (8.4)^2 (25.2)$$ $$= \pi \times 70.56 \times 25.2$$ $$= \pi \times 1777.31$$ 6. **Final volume (approximate):** $$V \approx 3.1416 \times 1777.31 = 5581.77 \text{ cm}^3$$ **Answer:** The volume of the cylinder is approximately $5581.77$ cubic centimeters.