The base radii of a cone and a cylinder are equal. If their curved surface areas are also equal, then the ratio of the slant height of the cone to the height of the cylinder is :

Options

A. `2 : 1`
B. `1 : 2`
C. `1 : 3`
D. `3 : 1`

Answer

To solve the problem, we need to find the ratio of the slant height of the cone to the height of the cylinder, given that the base radii and the curved surface areas (CSA) of both shapes are equal.

### Step-by-step Solution:

1. **Identify the Variables:**
Let:
- \( r \) = radius of the base of the cone and cylinder (since they are equal)
- \( L \) = slant height of the cone
- \( h \) = height of the cylinder

2. **Write the Formula for Curved Surface Area (CSA):**
- The CSA of the cone is given by the formula:
\[ \text{CSA}_{\text{cone}} = \pi r L \]
- The CSA of the cylinder is given by the formula:
\[ \text{CSA}_{\text{cylinder}} = 2 \pi r h \]

3. **Set the Curved Surface Areas Equal:**
Since the CSA of the cone is equal to the CSA of the cylinder, we can write:
\[ \pi r L = 2 \pi r h \]

4. **Cancel Out Common Terms:**
We can cancel \( \pi r \) from both sides (assuming \( r \neq 0 \)):
\[ L = 2h \]

5. **Find the Ratio of Slant Height to Height:**
We need to find the ratio of the slant height of the cone to the height of the cylinder:
\[ \frac{L}{h} = \frac{2h}{h} = 2 \]

6. **Final Result:**
Thus, the ratio of the slant height of the cone to the height of the cylinder is:
\[ \frac{L}{h} = 2 \]

### Conclusion:
The ratio of the slant height of the cone to the height of the cylinder is 2.