A cone of radius 4 cm is divided into two parts by drawing a plane through the mid-point of its axis and parallel to its base. Compare the volumes of the two parts.
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Answer

To solve the problem of comparing the volumes of the two parts of the cone, we will follow these steps:

### Step 1: Understand the cone's dimensions
Let the radius of the cone be \( r = 4 \) cm. The height of the cone is denoted as \( h \).

### Step 2: Divide the cone
The cone is divided into two parts by a plane through the midpoint of its axis, which means the height of the upper cone (let's call it \( h_1 \)) is \( \frac{h}{2} \) and the height of the lower frustum (let's call it \( h_2 \)) is also \( \frac{h}{2} \).

### Step 3: Determine the radius of the upper cone
Since the plane is drawn parallel to the base, the radius of the upper cone (let's denote it as \( r_1 \)) at the midpoint will be half of the original radius. Thus, \( r_1 = \frac{r}{2} = \frac{4}{2} = 2 \) cm.

### Step 4: Calculate the volume of the upper cone
The volume \( V_1 \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
For the upper cone:
\[ V_1 = \frac{1}{3} \pi (r_1^2) (h_1) = \frac{1}{3} \pi (2^2) \left(\frac{h}{2}\right) = \frac{1}{3} \pi (4) \left(\frac{h}{2}\right) \]
\[ V_1 = \frac{4\pi h}{6} = \frac{2\pi h}{3} \text{ cm}^3 \]

### Step 5: Calculate the volume of the lower frustum
To find the volume of the lower part (frustum), we need to use the formula for the volume of a frustum:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]
Where \( r_1 = 4 \) cm (base radius) and \( r_2 = 2 \) cm (top radius) and \( h = \frac{h}{2} \):
\[ V_2 = \frac{1}{3} \pi \left(\frac{h}{2}\right) \left(4^2 + 2^2 + 4 \cdot 2\right) \]
Calculating the terms inside the parentheses:
\[ 4^2 = 16, \quad 2^2 = 4, \quad 4 \cdot 2 = 8 \]
So,
\[ V_2 = \frac{1}{3} \pi \left(\frac{h}{2}\right) (16 + 4 + 8) = \frac{1}{3} \pi \left(\frac{h}{2}\right) (28) \]
\[ V_2 = \frac{28\pi h}{6} = \frac{14\pi h}{3} \text{ cm}^3 \]

### Step 6: Compare the volumes
Now we compare the volumes of the upper cone and the lower frustum:
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{\frac{2\pi h}{3}}{\frac{14\pi h}{3}} = \frac{2}{14} = \frac{1}{7} \]

### Conclusion
The ratio of the volume of the upper part (cone) to the volume of the lower part (frustum) is \( 1:7 \).