Question: A closed cylindrical water tank is to be painted black in color. The sum of the radius and height of the tank is $17\text{ m}$. The diameter of the tank is $6\text{ m}$. Find the total area painted.

1. **State the problem:** We need to find the total surface area of a closed cylindrical tank that will be painted black. 2. **Given:** - Sum of radius and height: $r + h = 17$ - Diameter of the tank: $6$ m, so radius $r = \frac{6}{2} = 3$ m 3. **Find:** Total surface area of the closed cylinder. 4. **Formula for total surface area of a closed cylinder:** $$A = 2\pi r^2 + 2\pi r h$$ where $2\pi r^2$ is the area of the two circular ends and $2\pi r h$ is the lateral surface area. 5. **Calculate height $h$:** $$r + h = 17 \implies 3 + h = 17 \implies h = 17 - 3 = 14$$ 6. **Calculate total surface area:** $$A = 2\pi (3)^2 + 2\pi (3)(14) = 2\pi (9) + 2\pi (42) = 18\pi + 84\pi = 102\pi$$ 7. **Final answer:** $$A = 102\pi \approx 102 \times 3.1416 = 320.44 \text{ m}^2$$ So, the total area painted is approximately $320.44$ square meters.