1. **State the problem:** We are given a cylinder with volume $178\pi$ cubic inches. We need to find the exact volume of a cone that has the same base area and height as this cylinder. 2. **Recall the formulas:** - Volume of a cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of a cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ 3. Since the cone and cylinder have the same base area and height, the values of $r$ and $h$ are the same for both. 4. From the cylinder volume, we have: $$178\pi = \pi r^2 h$$ Divide both sides by $\pi$: $$\cancel{\pi} 178 = \cancel{\pi} r^2 h$$ $$178 = r^2 h$$ 5. Substitute $r^2 h = 178$ into the cone volume formula: $$V_{cone} = \frac{1}{3} \pi (r^2 h) = \frac{1}{3} \pi (178) = \frac{178}{3} \pi$$ 6. **Final answer:** The exact volume of the cone is $$\frac{178}{3} \pi$$ cubic inches.