1. **State the problem:** We are given a cylinder with a radius $r = 5$ mm and a total surface area $S = 440$ square millimeters. We need to find the height $h$ of the cylinder. 2. **Formula for surface area of a cylinder:** $$S = 2\pi r^2 + 2\pi r h$$ This formula includes the areas of the two circular bases ($2\pi r^2$) and the rectangular side (lateral surface area, $2\pi r h$). 3. **Substitute known values:** $$440 = 2\pi (5)^2 + 2\pi (5) h$$ 4. **Calculate the area of the bases:** $$2\pi (5)^2 = 2\pi \times 25 = 50\pi$$ 5. **Rewrite the equation:** $$440 = 50\pi + 10\pi h$$ 6. **Isolate the term with $h$:** $$440 - 50\pi = 10\pi h$$ 7. **Divide both sides by $10\pi$ to solve for $h$:** $$h = \frac{440 - 50\pi}{10\pi}$$ 8. **Use the cancellation notation:** $$h = \frac{\cancel{10} \times 44 - 50\pi}{\cancel{10}\pi} = \frac{44 - 5\pi}{\pi}$$ 9. **Calculate numerical value:** Using $\pi \approx 3.1416$, $$h \approx \frac{44 - 5 \times 3.1416}{3.1416} = \frac{44 - 15.708}{3.1416} = \frac{28.292}{3.1416} \approx 9.01$$ 10. **Round to nearest whole number:** $$h \approx 9$$ millimeters. **Final answer:** The height of the cylinder is approximately 9 millimeters.