A right circular cone of radius 3 cm , has a curved surface area of `47.1 cm^(2)`. Find the volume of the cone , (use `pi=3.14`)

Answer

To find the volume of the cone, we first need to determine its height. We can use the formula for the curved surface area (CSA) of a cone, which is given by:

\[ \text{CSA} = \pi r l \]

where \( r \) is the radius and \( l \) is the slant height of the cone.

### Step 1: Use the curved surface area formula
Given:
- Curved Surface Area (CSA) = 47.1 cm²
- Radius \( r = 3 \) cm
- \( \pi = 3.14 \)

Substituting the values into the CSA formula:

\[ 47.1 = 3.14 \times 3 \times l \]

### Step 2: Solve for the slant height \( l \)
First, calculate \( 3.14 \times 3 \):

\[ 3.14 \times 3 = 9.42 \]

Now, substituting this back into the equation:

\[ 47.1 = 9.42 \times l \]

To find \( l \), divide both sides by 9.42:

\[ l = \frac{47.1}{9.42} \approx 5 \]

### Step 3: Use the Pythagorean theorem to find the height \( h \)
In a right circular cone, the radius, height, and slant height are related by the Pythagorean theorem:

\[ l^2 = r^2 + h^2 \]

Substituting the known values:

\[ 5^2 = 3^2 + h^2 \]

Calculating the squares:

\[ 25 = 9 + h^2 \]

Now, isolate \( h^2 \):

\[ h^2 = 25 - 9 = 16 \]

Taking the square root gives:

\[ h = \sqrt{16} = 4 \text{ cm} \]

### Step 4: Calculate the volume of the cone
The volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Substituting the known values:

\[ V = \frac{1}{3} \times 3.14 \times (3)^2 \times 4 \]

Calculating \( (3)^2 \):

\[ (3)^2 = 9 \]

Now substitute this back into the volume formula:

\[ V = \frac{1}{3} \times 3.14 \times 9 \times 4 \]

Calculating \( 9 \times 4 = 36 \):

\[ V = \frac{1}{3} \times 3.14 \times 36 \]

Calculating \( 3.14 \times 36 \):

\[ 3.14 \times 36 = 113.04 \]

Now divide by 3:

\[ V = \frac{113.04}{3} \approx 37.68 \text{ cm}^3 \]

### Final Answer
The volume of the cone is approximately \( 37.68 \text{ cm}^3 \).