Painting Cylindrical Tube 1F3420

1. **State the problem:** We need to find the cost of painting the outer surface of an open cylindrical tube (cup) given the sum of its radius and height, the diameter, and the cost per unit area. 2. **Given data:** - Diameter $$= 10\text{ cm}$$, so radius $$r = \frac{10}{2} = 5\text{ cm}$$. - Sum of radius and height $$r + h = 20$$, so height $$h = 20 - r = 20 - 5 = 15\text{ cm}$$. - Cost of painting $$\pi\text{ cm}^2$$ area is 3. 3. **Formula for outer surface area of an open cylinder:** $$\text{Surface Area} = \text{Curved Surface Area} + \text{Area of base} = 2\pi r h + \pi r^2$$ 4. **Calculate the surface area:** $$2\pi r h + \pi r^2 = \pi (2 r h + r^2) = \pi (2 \times 5 \times 15 + 5^2) = \pi (150 + 25) = \pi \times 175 = 175\pi\text{ cm}^2$$ 5. **Calculate the cost:** Cost per $$\pi\text{ cm}^2$$ is 3, so cost for $$175\pi\text{ cm}^2$$ is: $$3 \times 175 = 525$$ **Final answer:** The cost of painting the cup is 525.