The volume and heigh of a right circular cone are 1232 ` cm^(3)` and 24 cm respectively , the area of its curved surface ( in ` cm^(2)`) is :

Options

A. 1100
B. 225
C. 616
D. 550

Answer

To find the curved surface area of a right circular cone given its volume and height, we can follow these steps:

### Step 1: Write down the formulas
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

The curved surface area \( A \) of a cone is given by the formula:
\[ A = \pi r L \]
where \( L \) is the slant height of the cone.

### Step 2: Substitute the known values into the volume formula
Given:
- Volume \( V = 1232 \, \text{cm}^3 \)
- Height \( h = 24 \, \text{cm} \)

Substituting into the volume formula:
\[ 1232 = \frac{1}{3} \pi r^2 (24) \]

### Step 3: Simplify the equation
First, multiply both sides by 3 to eliminate the fraction:
\[ 1232 \times 3 = \pi r^2 (24) \]
\[ 3696 = \pi r^2 (24) \]

Next, divide both sides by \( 24 \):
\[ \frac{3696}{24} = \pi r^2 \]
Calculating the left side:
\[ 154 = \pi r^2 \]

### Step 4: Solve for \( r^2 \)
Now, divide both sides by \( \pi \):
\[ r^2 = \frac{154}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{154 \times 7}{22} = \frac{1078}{22} = 49 \]

### Step 5: Find the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 6: Calculate the slant height \( L \)
Using the Pythagorean theorem:
\[ L = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ L = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, \text{cm} \]

### Step 7: Calculate the curved surface area \( A \)
Now, substitute \( r \) and \( L \) into the curved surface area formula:
\[ A = \pi r L \]
Substituting the values:
\[ A = \frac{22}{7} \times 7 \times 25 \]
The \( 7 \) cancels out:
\[ A = 22 \times 25 = 550 \, \text{cm}^2 \]

### Final Answer
The area of the curved surface of the cone is:
\[ \boxed{550 \, \text{cm}^2} \]