A cone whose height is 15 cm and radius of base is 6 cm is trimmed suficiently to reduce it to a pyramid whose base is an quilateral triangle . The volume of the portion removed is :

Options

A. `330 cm^3`
B. `328 cm^(3) `
C. `325 cm^(3) `
D. `331 cm^(3)`

Answer

To solve the problem, we need to calculate the volume of the original cone and the volume of the pyramid formed after trimming. The volume of the portion removed will be the difference between these two volumes.

### Step 1: Calculate the volume of the cone

The formula for the volume \( V \) of a cone is given by:

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

Where:
- \( r \) is the radius of the base
- \( h \) is the height

Given:
- Radius \( r = 6 \) cm
- Height \( h = 15 \) cm

Substituting the values into the formula:

\[ V = \frac{1}{3} \pi (6)^2 (15) \]

Calculating \( (6)^2 \):

\[ (6)^2 = 36 \]

Now substituting back:

\[ V = \frac{1}{3} \pi (36)(15) \]

Calculating \( 36 \times 15 \):

\[ 36 \times 15 = 540 \]

So,

\[ V = \frac{1}{3} \pi (540) = 180\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the pyramid

The base of the pyramid is an equilateral triangle. The formula for the volume \( V \) of a pyramid is given by:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

#### Step 2.1: Calculate the area of the equilateral triangle

The formula for the area \( A \) of an equilateral triangle with side length \( a \) is:

\[ A = \frac{\sqrt{3}}{4} a^2 \]

To find the side length \( a \) of the equilateral triangle, we can use the radius of the circumcircle (which is the same as the radius of the cone's base). The radius \( r \) of the circumcircle of an equilateral triangle relates to its side length \( a \) as follows:

\[ r = \frac{a}{\sqrt{3}} \]

Given \( r = 6 \) cm, we can find \( a \):

\[ 6 = \frac{a}{\sqrt{3}} \implies a = 6\sqrt{3} \]

Now substituting \( a \) into the area formula:

\[ A = \frac{\sqrt{3}}{4} (6\sqrt{3})^2 \]

Calculating \( (6\sqrt{3})^2 \):

\[ (6\sqrt{3})^2 = 36 \times 3 = 108 \]

So,

\[ A = \frac{\sqrt{3}}{4} \times 108 = 27\sqrt{3} \text{ cm}^2 \]

#### Step 2.2: Calculate the volume of the pyramid

Now we need the height of the pyramid. Since the cone is trimmed to form the pyramid, we can assume the height of the pyramid is the same as the height of the cone, which is \( 15 \) cm.

Now substituting the area of the base and height into the pyramid volume formula:

\[ V = \frac{1}{3} \times (27\sqrt{3}) \times 15 \]

Calculating:

\[ V = \frac{1}{3} \times 405\sqrt{3} = 135\sqrt{3} \text{ cm}^3 \]

### Step 3: Calculate the volume of the portion removed

The volume of the portion removed is the volume of the cone minus the volume of the pyramid:

\[ \text{Volume removed} = V_{\text{cone}} - V_{\text{pyramid}} = 180\pi - 135\sqrt{3} \]

### Final Answer

The volume of the portion removed is:

\[ 180\pi - 135\sqrt{3} \text{ cm}^3 \]