1. **Stating the problem:** We are given the mantle area (lateral surface area) of a cylinder as 622 cm\textsuperscript{2} and the radius as 9 cm. We need to find the height of the cylinder. 2. **Formula used:** The lateral surface area (mantle) of a cylinder is given by the formula: $$A = 2 \pi r h$$ where $A$ is the lateral surface area, $r$ is the radius, and $h$ is the height. 3. **Substitute the known values:** $$622 = 2 \pi \times 9 \times h$$ 4. **Simplify the equation:** $$622 = 18 \pi h$$ 5. **Solve for $h$:** $$h = \frac{622}{18 \pi}$$ 6. **Show cancellation step:** $$h = \frac{\cancel{622}}{\cancel{18} \pi} \text{ (simplify numerator and denominator if possible)}$$ 7. **Calculate the numerical value:** Using $\pi \approx 3.1416$, $$h = \frac{622}{18 \times 3.1416} = \frac{622}{56.5488} \approx 11.0$$ **Final answer:** The height of the cylinder is approximately **11.0 cm**.