If a right circular cone of height 24 cm has a volume of 1232 `cm^3`. then the area of its curved surface (Take `pi=(22)/(7)`) is

Options

A. `1254 cm^2`
B. `704 cm^2`
C. `550 cm^2`
D. `154 cm^2`

Answer

To solve the problem of finding the curved surface area of a right circular cone with a height of 24 cm and a volume of 1232 cm³, we can follow these steps:

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

### Step 2: Substitute the known values into the volume formula
Given:
- Volume \( V = 1232 \, \text{cm}^3 \)
- Height \( h = 24 \, \text{cm} \)
- \( \pi = \frac{22}{7} \)

Substituting these values into the volume formula:
\[ 1232 = \frac{1}{3} \times \frac{22}{7} \times r^2 \times 24 \]

### Step 3: Simplify the equation
To simplify, first multiply both sides by 3:
\[ 3 \times 1232 = \frac{22}{7} \times r^2 \times 24 \]
\[ 3696 = \frac{22}{7} \times r^2 \times 24 \]

Next, multiply both sides by 7 to eliminate the fraction:
\[ 3696 \times 7 = 22 \times r^2 \times 24 \]
\[ 25872 = 22 \times r^2 \times 24 \]

Now, divide both sides by 24:
\[ \frac{25872}{24} = 22 \times r^2 \]
\[ 1078 = 22 \times r^2 \]

Finally, divide by 22 to solve for \( r^2 \):
\[ r^2 = \frac{1078}{22} \]
\[ r^2 = 49 \]

### Step 4: Calculate the radius
Taking the square root of both sides gives:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 5: Calculate the slant height \( l \)
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values:
\[ l = \sqrt{24^2 + 7^2} \]
\[ l = \sqrt{576 + 49} = \sqrt{625} = 25 \, \text{cm} \]

### Step 6: Calculate the curved surface area
The formula for the curved surface area \( A \) of a cone is:
\[ A = \pi r l \]
Substituting the values we have:
\[ A = \frac{22}{7} \times 7 \times 25 \]

### Step 7: Simplify the expression
The \( 7 \) in the numerator and denominator cancels out:
\[ A = 22 \times 25 = 550 \, \text{cm}^2 \]

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