1. **State the problem:** We need to find the radius $r$ of the base of a cone given the height $h=6$ inches and volume $V=464.7$ cubic inches. 2. **Given formula:** $$r=\left(\frac{3V}{\pi h}\right)^{\frac{1}{2}}$$ This means the radius is the square root of the fraction $\frac{3V}{\pi h}$. 3. **Substitute known values:** Use $\pi \approx 3.14$, $V=464.7$, and $h=6$: $$r=\left(\frac{3 \times 464.7}{3.14 \times 6}\right)^{\frac{1}{2}}$$ 4. **Calculate inside the parentheses:** $$\frac{3 \times 464.7}{3.14 \times 6} = \frac{1394.1}{18.84} \approx 73.98$$ 5. **Take the square root:** $$r = \sqrt{73.98} \approx 8.6$$ 6. **Final answer:** The radius of the cone's base is approximately **8.6 inches** rounded to 1 decimal place.