The ratio of curved surface areas of two cones is 1:8 and the ratio of their slant heights 1 :4. What is the ratio of the radii of the two cones ?

Options

A. a.`1:1`
B. b.`1:2`
C. c.`1:4`
D. d.`1:8`

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 slant heights.

### Step-by-Step Solution:

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

2. **Setting Up the Ratios**:
Let the radius and slant height of the first cone be \( r \) and \( l \) respectively, and for the second cone, let them be \( r_1 \) and \( l_1 \) respectively.

According to the problem:
- The ratio of the curved surface areas of the two cones is:
\[ \frac{\pi r l}{\pi r_1 l_1} = \frac{1}{8} \]
- The ratio of the slant heights is:
\[ \frac{l}{l_1} = \frac{1}{4} \]

3. **Substituting the Slant Height Ratio**:
From the slant height ratio, we can express \( l_1 \) in terms of \( l \):
\[ l_1 = 4l \]

4. **Substituting into the CSA Ratio**:
Now substitute \( l_1 \) into the CSA ratio:
\[ \frac{\pi r l}{\pi r_1 (4l)} = \frac{1}{8} \]
Simplifying this gives:
\[ \frac{r}{4r_1} = \frac{1}{8} \]

5. **Cross-Multiplying**:
Cross-multiplying to eliminate the fraction:
\[ 8r = 4r_1 \]

6. **Simplifying the Equation**:
Dividing both sides by 4:
\[ 2r = r_1 \]

7. **Finding the Ratio of the Radii**:
Rearranging gives:
\[ \frac{r}{r_1} = \frac{1}{2} \]

8. **Final Ratio**:
Therefore, the ratio of the radii of the two cones is:
\[ r : r_1 = 1 : 2 \]

### Final Answer:
The ratio of the radii of the two cones is \( 1 : 2 \).