A cone of radius 8cm and height 12cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of the two parts.

Answer

To find the ratio of the volumes of the two parts of the cone, we will follow these steps:

### Step 1: Calculate the volume of the original 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 of the cone.
- \( h \) is the height of the cone.

Given:
- Radius \( r = 8 \) cm
- Height \( h = 12 \) cm

Substituting the values into the formula:

\[ V = \frac{1}{3} \pi (8)^2 (12) \]

Calculating \( (8)^2 \):

\[ (8)^2 = 64 \]

Now substituting back:

\[ V = \frac{1}{3} \pi (64)(12) \]

Calculating \( 64 \times 12 \):

\[ 64 \times 12 = 768 \]

So,

\[ V = \frac{1}{3} \pi (768) = 256 \pi \text{ cm}^3 \]

### Step 2: Determine the dimensions of the smaller cone.
Since the cone is divided by a plane through the midpoint of its height, the height of the smaller cone will be half of the original height:

\[ \text{Height of smaller cone} = \frac{12}{2} = 6 \text{ cm} \]

The radius of the smaller cone will also be half of the original radius:

\[ \text{Radius of smaller cone} = \frac{8}{2} = 4 \text{ cm} \]

### Step 3: Calculate the volume of the smaller cone.
Using the same formula for the volume of a cone:

\[ V_{\text{small cone}} = \frac{1}{3} \pi r^2 h \]

Substituting the values for the smaller cone:

\[ V_{\text{small cone}} = \frac{1}{3} \pi (4)^2 (6) \]

Calculating \( (4)^2 \):

\[ (4)^2 = 16 \]

Now substituting back:

\[ V_{\text{small cone}} = \frac{1}{3} \pi (16)(6) \]

Calculating \( 16 \times 6 \):

\[ 16 \times 6 = 96 \]

So,

\[ V_{\text{small cone}} = \frac{1}{3} \pi (96) = 32 \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the frustum.
The volume of the frustum is the volume of the original cone minus the volume of the smaller cone:

\[ V_{\text{frustum}} = V_{\text{original}} - V_{\text{small cone}} = 256 \pi - 32 \pi = 224 \pi \text{ cm}^3 \]

### Step 5: Find the ratio of the volumes of the two parts.
The ratio of the volume of the frustum to the volume of the smaller cone is:

\[ \text{Ratio} = \frac{V_{\text{frustum}}}{V_{\text{small cone}}} = \frac{224 \pi}{32 \pi} \]

The \( \pi \) cancels out:

\[ \text{Ratio} = \frac{224}{32} = 7 \]

Thus, the ratio of the volumes of the two parts is:

\[ \text{Ratio} = 7:1 \]

### Final Answer:
The ratio of the volumes of the two parts is \( 7:1 \).

---