The curved surface area of a right circular cone is `156pi` and the radius of its base is 12 cm. What is the volume of the cone, in `"cm"^(3)` ?

Options

A. `192pi`
B. `210pi`
C. `240pi`
D. `180pi`

Answer

To find the volume of the cone, we will follow these steps:

### Step 1: Understand the formula for the curved surface area of a cone.
The curved surface area (CSA) of a right circular 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 2: Substitute the known values into the CSA formula.
We are given that the CSA is \( 156\pi \) and the radius \( R \) is \( 12 \) cm. Substituting these values into the formula:
\[ 156\pi = \pi \cdot 12 \cdot L \]

### Step 3: Simplify the equation.
We can cancel \( \pi \) from both sides:
\[ 156 = 12L \]

### Step 4: Solve for the slant height \( L \).
Now, divide both sides by \( 12 \):
\[ L = \frac{156}{12} = 13 \text{ cm} \]

### Step 5: Use the Pythagorean theorem to find the height \( H \) of the cone.
In a right circular cone, the height \( H \), radius \( R \), and slant height \( L \) are related by the Pythagorean theorem:
\[ L^2 = R^2 + H^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + H^2 \]
\[ 169 = 144 + H^2 \]

### Step 6: Solve for \( H^2 \).
Subtract \( 144 \) from both sides:
\[ H^2 = 169 - 144 = 25 \]

### Step 7: Find the height \( H \).
Taking the square root of both sides:
\[ H = \sqrt{25} = 5 \text{ cm} \]

### Step 8: Calculate the volume of the cone.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi R^2 H \]
Substituting the values of \( R \) and \( H \):
\[ V = \frac{1}{3} \pi (12^2) (5) \]
\[ = \frac{1}{3} \pi (144) (5) \]
\[ = \frac{1}{3} \pi (720) \]
\[ = 240\pi \text{ cm}^3 \]

### Final Answer:
The volume of the cone is \( 240\pi \text{ cm}^3 \).

---