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. Find the ratio of their radii.

Answer

To solve the problem, we need to find the ratio of the radii of two cones given certain conditions about their curved surface areas and slant heights. Here’s a step-by-step solution:

### Step 1: Define the Variables
Let:
- \( R_1 \) = radius of the first cone
- \( R_2 \) = radius of the second cone
- \( L_1 \) = slant height of the first cone
- \( L_2 \) = slant height of the second cone

### Step 2: Write Down the Given Information
From the problem, we know:
1. The curved surface area (CSA) of the first cone is twice that of the second cone:
\[ CSA_1 = 2 \times CSA_2 \]
2. The slant height of the second cone is twice that of the first cone:
\[ L_2 = 2 \times L_1 \]

### Step 3: Use the Formula for Curved Surface Area of a Cone
The formula for the curved surface area of a cone is given by:
\[ CSA = \pi R L \]
Using this formula, we can express the CSA for both cones:
- For the first cone:
\[ CSA_1 = \pi R_1 L_1 \]
- For the second cone:
\[ CSA_2 = \pi R_2 L_2 \]

### Step 4: Set Up the Equation
Substituting the expressions for CSA into the given relationship:
\[ \pi R_1 L_1 = 2 \times \pi R_2 L_2 \]
Cancelling \( \pi \) from both sides, we have:
\[ R_1 L_1 = 2 R_2 L_2 \]

### Step 5: Substitute the Value of \( L_2 \)
Now, substitute \( L_2 = 2 L_1 \) into the equation:
\[ R_1 L_1 = 2 R_2 (2 L_1) \]
This simplifies to:
\[ R_1 L_1 = 4 R_2 L_1 \]

### Step 6: Cancel \( L_1 \) from Both Sides
Assuming \( L_1 \neq 0 \), we can divide both sides by \( L_1 \):
\[ R_1 = 4 R_2 \]

### Step 7: Find the Ratio of the Radii
From the equation \( R_1 = 4 R_2 \), we can express the ratio of the radii:
\[ \frac{R_1}{R_2} = 4 \]
Thus, the ratio of their radii is:
\[ R_1 : R_2 = 4 : 1 \]

### Final Answer
Hence, the required ratio of the radii of the two cones is \( 4 : 1 \).

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