There are two cones. The curved surface area of one is twice that of the other. The slant height of the latter is twice that of the former. The ratio of their radii is

Options

A. `4:1`
B. `4:3`
C. `3:4`
D. `1:4`

Answer

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

### Step-by-Step Solution:

1. **Understand the Curved Surface Area (CSA) Formula**:
The formula for the curved surface area of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

2. **Set Up the Equations**:
Let the first cone have radius \( r_1 \) and slant height \( l_1 \). The second cone has radius \( r_2 \) and slant height \( l_2 \).
According to the problem:
- The curved surface area of the first cone is twice that of the second cone:
\[ \pi r_1 l_1 = 2 \pi r_2 l_2 \]
- The slant height of the second cone is twice that of the first cone:
\[ l_2 = 2 l_1 \]

3. **Substitute the Slant Height**:
Substitute \( l_2 = 2 l_1 \) into the CSA equation:
\[ \pi r_1 l_1 = 2 \pi r_2 (2 l_1) \]
Simplifying this gives:
\[ \pi r_1 l_1 = 4 \pi r_2 l_1 \]

4. **Cancel Common Terms**:
Since \( \pi \) and \( l_1 \) are common on both sides, we can cancel them out (assuming \( l_1 \neq 0 \)):
\[ r_1 = 4 r_2 \]

5. **Find the Ratio of the Radii**:
From the equation \( r_1 = 4 r_2 \), we can express the ratio of the radii as:
\[ \frac{r_1}{r_2} = 4 \]
Thus, the ratio of the radii \( r_1 : r_2 \) is:
\[ r_1 : r_2 = 4 : 1 \]

### Final Answer:
The ratio of their radii is \( 4 : 1 \).

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