The ratio of curved surface area of two cones is 1:4 and the ratio of slant height of the two cones is 2 : 1. What is the ratio of the radius of the two cones ?

Options

A. `1:2`
B. `1:4`
C. `1:8`
D. `1:1`

Answer

To find the ratio of the radius of the two cones given the ratio of their curved surface areas and slant heights, we can follow these steps:

### Step 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 and \( l \) is the slant height of the cone.

### Step 2: Set up the ratios
Let the radius and slant height of the first cone be \( r_1 \) and \( l_1 \), and for the second cone, let them be \( r_2 \) and \( l_2 \). We are given:
- The ratio of the curved surface areas of the two cones is \( \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{1}{4} \).
- The ratio of the slant heights of the two cones is \( \frac{l_1}{l_2} = \frac{2}{1} \).

### Step 3: Express the curved surface areas
Using the formula for CSA, we can write:
\[ \text{CSA}_1 = \pi r_1 l_1 \]
\[ \text{CSA}_2 = \pi r_2 l_2 \]
Thus, the ratio of the curved surface areas can be expressed as:
\[ \frac{\pi r_1 l_1}{\pi r_2 l_2} = \frac{1}{4} \]
This simplifies to:
\[ \frac{r_1 l_1}{r_2 l_2} = \frac{1}{4} \]

### Step 4: Substitute the ratio of slant heights
From the ratio of slant heights, we know:
\[ l_1 = 2l_2 \]
Substituting this into the curved surface area ratio gives:
\[ \frac{r_1 (2l_2)}{r_2 l_2} = \frac{1}{4} \]
The \( l_2 \) terms cancel out:
\[ \frac{2r_1}{r_2} = \frac{1}{4} \]

### Step 5: Solve for the ratio of the radii
Rearranging the equation gives:
\[ 2r_1 = \frac{1}{4} r_2 \]
Multiplying both sides by 4:
\[ 8r_1 = r_2 \]
Thus, the ratio of the radii is:
\[ \frac{r_1}{r_2} = \frac{1}{8} \]

### Final Answer
The ratio of the radius of the two cones is \( 1:8 \).

---