If the curved surface area of a cone is thrice that of another cone and slant height of the second cone is thrice that of the first, find the ratio of the area of their base.

Options

A. `81 : 1`
B. `9 : 1`
C. `3 : 1`
D. `27 : 1`

Answer

To solve the problem, we need to find the ratio of the areas of the bases of two cones given certain conditions about their curved surface areas and slant heights.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The curved surface area (CSA) of a cone is given by the formula:
\[ \text{CSA} = \pi R L \]
where \( R \) is the radius of the base and \( L \) is the slant height.

2. **Set Up the Given Conditions**:
- Let the first cone have a radius \( r \) and slant height \( l \).
- Let the second cone have a radius \( R \) and slant height \( L \).
- According to the problem:
- The curved surface area of the second cone is three times that of the first cone:
\[ \pi R L = 3 \cdot \pi r l \]
- The slant height of the second cone is three times that of the first cone:
\[ L = 3l \]

3. **Substitute the Slant Height**:
- Substitute \( L = 3l \) into the CSA equation:
\[ \pi R (3l) = 3 \cdot \pi r l \]
- This simplifies to:
\[ 3\pi R l = 3\pi r l \]
- Dividing both sides by \( 3\pi l \) (assuming \( l \neq 0 \)):
\[ R = r \]

4. **Relate the Radii**:
- From the previous step, we have \( R = r \). However, we need to find the relationship between the areas of the bases.
- We know from the problem that the CSA of the first cone is three times that of the second cone, which gives us:
\[ \pi R L = 3 \cdot \pi r l \]

5. **Substituting Again**:
- Substitute \( L = 3l \) again into the CSA equation:
\[ \pi R (3l) = 3 \cdot \pi r l \]
- Simplifying gives:
\[ 3\pi R l = 3\pi r l \]
- Dividing both sides by \( 3\pi l \):
\[ R = 3r \]

6. **Find the Ratio of the Areas of the Bases**:
- The area of the base of the first cone is:
\[ A_1 = \pi r^2 \]
- The area of the base of the second cone is:
\[ A_2 = \pi R^2 = \pi (3r)^2 = 9\pi r^2 \]
- Now, find the ratio of the areas:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\pi r^2}{9\pi r^2} = \frac{1}{9} \]

7. **Final Ratio**:
- Thus, the ratio of the area of the base of the first cone to the second cone is:
\[ 1:9 \]

### Conclusion:
The ratio of the area of their bases is \( 1:9 \).