1. **Problem:** Find the radius of a cylinder with height $27$ mm and volume $4300$ mm$^3$. 2. **Formula:** The volume $V$ of a cylinder is given by $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Substitute known values:** $$4300 = \pi r^2 \times 27$$ 4. **Solve for $r^2$:** $$r^2 = \frac{4300}{27\pi}$$ 5. **Calculate $r^2$:** Using $\pi \approx 3.1416$, $$r^2 = \frac{4300}{27 \times 3.1416} \approx \frac{4300}{84.8232} \approx 50.69$$ 6. **Find $r$ by taking the square root:** $$r = \sqrt{50.69} \approx 7.12 \text{ mm}$$ 7. **Answer:** The radius of the cylinder is approximately **7.12 mm** to 2 decimal places.