If the curved surface area of a cone is thrice that of another cone and slant height of the second cone is thrice that of the first, find the ratio of the area of their base.

Options

A. A)`81:1`
B. B)`9:1`
C. C)` 3:1`
D. D)`27:1`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We have two cones. The curved surface area (CSA) of the first cone is three times that of the second cone. Additionally, the slant height of the second cone is three times that of the first cone.

### Step 2: 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 of the base and \( l \) is the slant height.

### Step 3: Set up the equations based on the given information
Let:
- \( r_1 \) and \( l_1 \) be the radius and slant height of the first cone.
- \( r_2 \) and \( l_2 \) be the radius and slant height of the second cone.

From the problem, we have:
1. \( \text{CSA of first cone} = 3 \times \text{CSA of second cone} \)
\[ \pi r_1 l_1 = 3(\pi r_2 l_2) \]
Simplifying this, we get:
\[ r_1 l_1 = 3 r_2 l_2 \quad \text{(Equation 1)} \]

2. The slant height of the second cone is three times that of the first:
\[ l_2 = 3 l_1 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Substituting \( l_2 = 3 l_1 \) into Equation 1:
\[ r_1 l_1 = 3 r_2 (3 l_1) \]
This simplifies to:
\[ r_1 l_1 = 9 r_2 l_1 \]
Assuming \( l_1 \neq 0 \), we can divide both sides by \( l_1 \):
\[ r_1 = 9 r_2 \quad \text{(Equation 3)} \]

### Step 5: Calculate the ratio of the areas of the bases
The area of the base of a cone is given by:
\[ \text{Area} = \pi r^2 \]
Thus, the areas of the bases of the two cones are:
- Area of the base of the first cone: \( A_1 = \pi r_1^2 \)
- Area of the base of the second cone: \( A_2 = \pi r_2^2 \)

Now, we need to find the ratio \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{r_1^2}{r_2^2} \]

### Step 6: Substitute Equation 3 into the ratio
From Equation 3, we know \( r_1 = 9 r_2 \):
\[ \frac{A_1}{A_2} = \frac{(9 r_2)^2}{r_2^2} = \frac{81 r_2^2}{r_2^2} = 81 \]

### Final Result
Thus, the ratio of the area of their bases is:
\[ \frac{A_1}{A_2} = 81 : 1 \]