1. **State the problem:** We need to find the slant height $l$ of a cone given its surface area $S = 204.2$ and radius $r = 5$. 2. **Recall the formula for the surface area of a cone:** $$ S = \pi r^2 + \pi r l $$ where $r$ is the radius and $l$ is the slant height. 3. **Substitute the known values:** $$ 204.2 = \pi \times 5^2 + \pi \times 5 \times l $$ 4. **Simplify the equation:** $$ 204.2 = 25\pi + 5\pi l $$ 5. **Isolate $l$:** $$ 204.2 - 25\pi = 5\pi l $$ $$ l = \frac{204.2 - 25\pi}{5\pi} $$ 6. **Calculate the numerical value:** Using $\pi \approx 3.1416$, $$ 25\pi \approx 78.54 $$ $$ 204.2 - 78.54 = 125.66 $$ $$ 5\pi \approx 15.708 $$ $$ l = \frac{125.66}{15.708} \approx 8.0 $$ **Final answer:** The slant height $l$ is approximately $8.0$ units.