A and B are two cones. The curved surface area of A is twice that of B. The slant height of B is twice that of A. Whatis the ratio of radii of A to B?

Options

A. 3:2
B. 1:4
C. 4:1
D. 2:1

Answer

To solve the problem, we need to find the ratio of the radii of two cones, A and B, given the following information:

1. The curved surface area of cone A is twice that of cone B.
2. The slant height of cone B is twice that of cone A.

Let's denote:
- \( r_1 \) = radius of cone A
- \( r_2 \) = radius of cone B
- \( l_1 \) = slant height of cone A
- \( l_2 \) = slant height of cone B

### Step 1: Write the formula for the 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.

### Step 2: Set up the equations based on the given information.
From the problem statement, we have:
\[ \text{CSA of A} = 2 \times \text{CSA of B \]
This can be expressed as:
\[ \pi r_1 l_1 = 2 \times \pi r_2 l_2 \]

### Step 3: Substitute the relationship between the slant heights.
We know that the slant height of cone B is twice that of cone A:
\[ l_2 = 2 l_1 \]
Substituting this into the CSA equation gives:
\[ \pi r_1 l_1 = 2 \times \pi r_2 (2 l_1) \]
This simplifies to:
\[ \pi r_1 l_1 = 4 \pi r_2 l_1 \]

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

### Step 5: Find the ratio of the radii.
To find the ratio of the radii \( r_1 \) to \( r_2 \):
\[ \frac{r_1}{r_2} = 4 \]
Thus, the ratio of the radius of cone A to the radius of cone B is:
\[ \frac{r_1}{r_2} = 4:1 \]

### Final Answer:
The ratio of the radii of cone A to cone B is \( 4:1 \).

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