A solid cylinder of base radius 7 cm and height 24 cm is surmounted by a cone of the same radius and same vertical height. A hemisphere surmounts the cylinder at the other end. Surface area of the solid will be

### Concept & Composite Surface Area For a composite solid formed by joining shapes, only the exposed outer surfaces are counted. The flat bases where the shapes touch are hidden and excluded. Total Surface Area = $$CSA_{cone} + CSA_{cylinder} + CSA_{hemisphere}$$ $$= \pi r l + 2\pi r h + 2\pi r^2$$ ### Step-by-Step Solution 1. Identify the given parameters: $r = 7$ cm, $h = 24$ cm. 2. First, calculate the slant height ($l$) of the cone: $l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$ cm. 3. Calculate the exposed curved surface area of the cone: $CSA_{cone} = \pi r l = \pi (7)(25) = 175\pi$. 4. Calculate the exposed curved surface area of the middle cylinder: $CSA_{cylinder} = 2\pi r h = 2\pi (7)(24) = 336\pi$. 5. Calculate the exposed curved surface area of the bottom hemisphere: $CSA_{hemisphere} = 2\pi r^2 = 2\pi (7)^2 = 2\pi (49) = 98\pi$. 6. Sum the areas to find the total surface area: Total SA $= 175\pi + 336\pi + 98\pi = 609\pi$ cm$^2$. ### Exam Strategy & Shortcut You can factor out $\pi r$ from the entire expression to streamline the arithmetic: $\pi r (l + 2h + 2r)$. Substitute: $\pi(7) \times (25 + 48 + 14) = 7\pi \times 87$. Multiply $7 \times 87$: $7 \times 80 = 560$ and $7 \times 7 = 49$. $560 + 49 = 609$. Hence, $609\pi$. ### Common Pitfall A classic mistake is adding the flat circular base areas ($\pi r^2$) into the surface area calculation. Remember that when shapes "surmount" each other, the joining faces become internal and are no longer part of the surface area. ### Final Answer Therefore, the correct answer is **$609\pi$ cm$^2$**.