1. **Problem statement:** A cone and a cylinder have the same base radius and volume. Find the ratio of the height of the cone to the height of the cylinder. 2. **Formula for volume:** - Volume of a cylinder: $$V_{cyl} = \pi r^2 h_{cyl}$$ - Volume of a cone: $$V_{cone} = \frac{1}{3} \pi r^2 h_{cone}$$ 3. **Given:** Same base radius $r$ and equal volumes, so $$V_{cone} = V_{cyl}$$ 4. **Set volumes equal:** $$\frac{1}{3} \pi r^2 h_{cone} = \pi r^2 h_{cyl}$$ 5. **Simplify by dividing both sides by $\pi r^2$ (non-zero):** $$\frac{1}{3} h_{cone} = h_{cyl}$$ 6. **Solve for the ratio:** $$\frac{h_{cone}}{h_{cyl}} = 3$$ **Final answer:** The ratio of the height of the cone to the height of the cylinder is $3$.