1. **State the problem:** We are given the height $h=3$ cm and the volume $V=8478$ cm$^3$ of a cylinder, and we need to find the radius $r$. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Rearrange the formula to solve for $r^2$:** $$r^2 = \frac{V}{\pi h}$$ 4. **Substitute the known values:** $$r^2 = \frac{8478}{\pi \times 3}$$ 5. **Calculate the denominator:** $$\pi \times 3 \approx 3.1416 \times 3 = 9.4248$$ 6. **Divide to find $r^2$:** $$r^2 = \frac{8478}{9.4248} \approx 899.7$$ 7. **Take the square root to find $r$:** $$r = \sqrt{899.7} \approx 29.99$$ 8. **Final answer:** The radius of the cylinder is approximately $30$ cm.