The radii of two cones are equal and their slant heights are in the ratio 3:2. If the curved surface area of the smaller cone is ` 300 cm^(2)`, then the curved surface area of the bigger cone ( in ` cm^2`) is :

Options

A. 250
B. 450
C. 150
D. 200

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Given Information
We have two cones with equal radii. The slant heights of the cones are in the ratio of 3:2. The curved surface area of the smaller cone is given as 300 cm².

### Step 2: Write the Formula for Curved Surface Area
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 3: Set Up the Ratios
Let:
- \( r \) = radius of both cones (equal for both)
- \( l_1 \) = slant height of the bigger cone
- \( l_2 \) = slant height of the smaller cone

From the problem, we know:
\[ \frac{l_1}{l_2} = \frac{3}{2} \]
This implies:
\[ l_1 = 3x \quad \text{and} \quad l_2 = 2x \]
for some value \( x \).

### Step 4: Calculate the Radius Using the Smaller Cone's CSA
Using the CSA of the smaller cone:
\[ \text{CSA}_{\text{smaller}} = \pi r l_2 = 300 \]
Substituting \( l_2 \):
\[ \pi r (2x) = 300 \]
This simplifies to:
\[ 2\pi r x = 300 \]
\[ \pi r x = 150 \]
Thus, we can express \( r \) in terms of \( x \):
\[ r = \frac{150}{\pi x} \]

### Step 5: Calculate the Curved Surface Area of the Bigger Cone
Now, we can find the CSA of the bigger cone:
\[ \text{CSA}_{\text{bigger}} = \pi r l_1 \]
Substituting \( l_1 \):
\[ \text{CSA}_{\text{bigger}} = \pi r (3x) \]
Now substituting the value of \( r \):
\[ \text{CSA}_{\text{bigger}} = \pi \left(\frac{150}{\pi x}\right) (3x) \]
The \( \pi \) and \( x \) cancel out:
\[ \text{CSA}_{\text{bigger}} = 150 \times 3 = 450 \, \text{cm}^2 \]

### Conclusion
The curved surface area of the bigger cone is \( 450 \, \text{cm}^2 \).

---