The ratio of curved surface area of two cones is 1:9 and the ratio of slant height of the two cones is 3 :1. What is the ratio of the radius of the two cones ?

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two cones. Let’s denote the first cone as Cone 1 and the second cone as Cone 2.
- The ratio of their curved surface areas is given as \(1:9\).
- The ratio of their slant heights is given as \(3:1\).

2. **Formulating the Curved Surface Area**:
- The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \(r\) is the radius and \(l\) is the slant height.
- Let the radius of Cone 1 be \(R_1\) and the slant height be \(L_1\).
- Let the radius of Cone 2 be \(R_2\) and the slant height be \(L_2\).

3. **Using the Given Ratios**:
- From the problem, we know:
\[ \frac{\text{CSA of Cone 1}}{\text{CSA of Cone 2}} = \frac{1}{9} \]
This can be expressed as:
\[ \frac{\pi R_1 L_1}{\pi R_2 L_2} = \frac{1}{9} \]
Simplifying this gives:
\[ \frac{R_1 L_1}{R_2 L_2} = \frac{1}{9} \quad \text{(Equation 1)} \]

4. **Substituting the Slant Height Ratio**:
- We are also given the ratio of the slant heights:
\[ \frac{L_1}{L_2} = \frac{3}{1} \]
This means \(L_1 = 3L_2\).

5. **Substituting into Equation 1**:
- Now, substituting \(L_1\) in Equation 1:
\[ \frac{R_1 (3L_2)}{R_2 L_2} = \frac{1}{9} \]
- Cancelling \(L_2\) from both sides (assuming \(L_2 \neq 0\)):
\[ \frac{3R_1}{R_2} = \frac{1}{9} \]

6. **Cross Multiplying**:
- Cross multiplying gives:
\[ 3R_1 = \frac{1}{9} R_2 \]
- Rearranging this gives:
\[ R_1 = \frac{1}{27} R_2 \]

7. **Finding the Ratio of the Radii**:
- Therefore, the ratio of the radii \(R_1 : R_2\) is:
\[ R_1 : R_2 = \frac{1}{27} : 1 = 1 : 27 \]

### Final Answer:
The ratio of the radius of the two cones is \(1:27\).

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