A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surfaces will be

### Concept & Geometry Integration To find the ratio of their curved surfaces, we first need to express the dimensions of both shapes in terms of a single variable. For a hemisphere, its height is inherently equal to its base radius. $$ \text{Hemisphere CSA} = 2\pi r^2 $$ $$ \text{Cone CSA} = \pi rl $$ (where $l = \sqrt{r^2 + h^2}$) ### Step-by-Step Solution 1. **Establish Dimensions:** - Equal bases mean they share the same radius, $r$. - The height of a hemisphere is equal to its radius ($r$). - Since their heights are equal, the height of the cone ($h$) is also $r$. ($h = r$) 2. **Calculate Slant Height of Cone ($l$):** - $l = \sqrt{r^2 + h^2}$ - Substitute $h = r$: $l = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2}$ 3. **Calculate Both Curved Surface Areas:** - Hemisphere $\text{CSA} = 2\pi r^2$ - Cone $\text{CSA} = \pi r (r\sqrt{2}) = \sqrt{2}\pi r^2$ 4. **Find the Ratio:** - $\frac{\text{Hemisphere CSA}}{\text{Cone CSA}} = \frac{2\pi r^2}{\sqrt{2}\pi r^2}$ - Ratio $= \frac{2}{\sqrt{2}} = \frac{\sqrt{2} \times \sqrt{2}}{\sqrt{2}} = \sqrt{2}$ - Ratio $= \sqrt{2} : 1$ ### Exam Strategy & Shortcut Recognize that a cone with equal height and radius forms an isosceles right triangle in cross-section. Its slant edge ratio is $\sqrt{2}$. The hemisphere's CSA multiplier is $2$, and the cone's is $\sqrt{2}$. The ratio $\frac{2}{\sqrt{2}}$ simplifies to $\sqrt{2}:1$ directly. ### Common Pitfall A common misstep is failing to realize that the height of a hemisphere is strictly equal to its radius. Without this deduction, the problem appears unsolvable due to missing information. ### Final Answer Therefore, the correct answer is **$\sqrt{2} : 1$**.