Question: The total surface area of a closed cylindrical tank is $$102\pi\text{ m}^2$$. The sum of the radius and height of the tank is $$17\text{ m}$$. Find the radius of the base of the tank.
1. **State the problem:** We need to find the radius $$r$$ of a closed cylindrical tank given that its total surface area is $$102\pi$$ square meters and the sum of its radius and height $$h$$ is $$17$$ meters.
2. **Recall the formula for the total surface area of a closed cylinder:**
$$\text{Surface Area} = 2\pi r^2 + 2\pi r h$$
This includes the areas of the two circular bases and the curved surface.
3. **Given:**
$$2\pi r^2 + 2\pi r h = 102\pi$$
and
$$r + h = 17$$
4. **Express height $$h$$ in terms of radius $$r$$:**
$$h = 17 - r$$
5. **Substitute $$h$$ into the surface area equation:**
$$2\pi r^2 + 2\pi r (17 - r) = 102\pi$$
6. **Divide both sides by $$2\pi$$ to simplify:**
$$r^2 + r(17 - r) = 51$$
7. **Expand and simplify:**
$$r^2 + 17r - r^2 = 51$$
$$17r = 51$$
8. **Solve for $$r$$:**
$$r = \frac{51}{17}$$
9. **Simplify the fraction:**
$$r = 3$$
**Final answer:** The radius of the base of the tank is $$3\text{ m}$$.