The volume of a right circular cone of height 24 cm is `1232 cm^(3)`. Its curved surface area is

Options

A. `1254 cm^(2)`
B. `704 cm^(2)`
C. `550 cm^(2)`
D. `462 cm^(2)`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Height (h) of the cone = 24 cm
- Volume (V) of the cone = 1232 cm³

2. **Use the Volume Formula for a Cone:**
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Here, \( \pi \) can be approximated as \( \frac{22}{7} \).

3. **Substitute the Known Values into the Formula:**
Substitute \( V = 1232 \, \text{cm}^3 \) and \( h = 24 \, \text{cm} \) into the volume formula:
\[ 1232 = \frac{1}{3} \times \frac{22}{7} \times r^2 \times 24 \]

4. **Rearranging the Equation to Solve for \( r^2 \):**
Multiply both sides by 3 to eliminate the fraction:
\[ 1232 \times 3 = \frac{22}{7} \times r^2 \times 24 \]
\[ 3696 = \frac{22 \times r^2 \times 24}{7} \]
Now, multiply both sides by 7:
\[ 3696 \times 7 = 22 \times r^2 \times 24 \]
\[ 25872 = 22 \times r^2 \times 24 \]
Divide both sides by 24:
\[ \frac{25872}{24} = 22 \times r^2 \]
\[ 1078 = 22 \times r^2 \]
Finally, divide by 22 to find \( r^2 \):
\[ r^2 = \frac{1078}{22} = 49 \]

5. **Calculate the Radius \( r \):**
Now, take the square root of \( r^2 \):
\[ r = \sqrt{49} = 7 \, \text{cm} \]

6. **Calculate the Slant Height \( l \):**
Use the Pythagorean theorem to find the slant height \( l \):
\[ l = \sqrt{h^2 + r^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \, \text{cm} \]

7. **Use the Curved Surface Area Formula:**
The formula for the curved surface area (CSA) of a cone is:
\[ \text{CSA} = \pi r l \]
Substitute \( r = 7 \, \text{cm} \) and \( l = 25 \, \text{cm} \):
\[ \text{CSA} = \frac{22}{7} \times 7 \times 25 \]

8. **Calculate the Curved Surface Area:**
Simplifying:
\[ \text{CSA} = 22 \times 25 = 550 \, \text{cm}^2 \]

### Final Answer:
The curved surface area of the cone is \( 550 \, \text{cm}^2 \).

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