Question: A cylinder has a height of $5$ mm and a radius of $9$ mm. What is the surface area of the cylinder? Use $3.14$ or a calculator button for $\pi$ and provide an answer accurate to the nearest hundredth.
1. **State the problem:** We need to find the surface area of a cylinder with height $h=5$ mm and radius $r=9$ mm.
2. **Formula for surface area of a cylinder:**
$$\text{Surface Area} = 2\pi r^2 + 2\pi rh$$
This formula includes the areas of the two circular bases ($2\pi r^2$) and the rectangular side (lateral surface area, $2\pi rh$).
3. **Substitute the values:**
$$2 \times 3.14 \times 9^2 + 2 \times 3.14 \times 9 \times 5$$
4. **Calculate each part:**
$$2 \times 3.14 \times 81 + 2 \times 3.14 \times 45$$
$$= 2 \times 3.14 \times 81 + 2 \times 3.14 \times 45$$
5. **Simplify:**
$$= 2 \times 254.34 + 2 \times 141.3$$
$$= 508.68 + 282.6$$
6. **Add to get total surface area:**
$$508.68 + 282.6 = 791.28$$
7. **Final answer:** The surface area of the cylinder is approximately $791.28$ square millimeters.