If a cone is cut into two parts by a horizontal plane passing through the mid point of its axis , the ratio of the volumes of the upper part and the frustum is :

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `1:7`

Answer

To solve the problem of finding the ratio of the volumes of the upper part of a cone (the smaller cone) and the frustum (the remaining part of the cone after cutting), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Problem**: We have a cone that is cut horizontally at the midpoint of its height. We need to find the ratio of the volume of the upper part (the smaller cone) to the volume of the frustum (the lower part).

2. **Define the Dimensions of the Cone**:
- Let the height of the original cone be \( h \).
- The radius of the base of the cone be \( r \).

3. **Calculate the Volume of the Original Cone**:
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
For simplicity, let’s assume \( h = 1 \) and \( r = 1 \). Thus, the volume of the original cone becomes:
\[ V = \frac{1}{3} \pi (1^2)(1) = \frac{1}{3} \pi \]

4. **Determine the Height of the Smaller Cone**:
Since the cone is cut at the midpoint, the height of the smaller cone (upper part) is:
\[ h_1 = \frac{h}{2} = \frac{1}{2} \]

5. **Calculate the Radius of the Smaller Cone**:
The radius of the smaller cone is also halved because the dimensions are proportional. Thus:
\[ r_1 = \frac{r}{2} = \frac{1}{2} \]

6. **Calculate the Volume of the Smaller Cone**:
Using the volume formula for the smaller cone:
\[ V_1 = \frac{1}{3} \pi r_1^2 h_1 = \frac{1}{3} \pi \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right) = \frac{1}{3} \pi \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{24} \pi \]

7. **Calculate the Volume of the Frustum**:
The volume of the frustum \( V_3 \) can be found by subtracting the volume of the smaller cone from the volume of the original cone:
\[ V_3 = V - V_1 = \frac{1}{3} \pi - \frac{1}{24} \pi \]
To perform this subtraction, we need a common denominator:
\[ V = \frac{8}{24} \pi \quad \text{(converting to 24 as the denominator)} \]
Thus:
\[ V_3 = \frac{8}{24} \pi - \frac{1}{24} \pi = \frac{7}{24} \pi \]

8. **Find the Ratio of the Volumes**:
Now, we find the ratio of the volume of the upper part (smaller cone) to the volume of the frustum:
\[ \text{Ratio} = \frac{V_1}{V_3} = \frac{\frac{1}{24} \pi}{\frac{7}{24} \pi} = \frac{1}{7} \]

### Final Answer
The ratio of the volumes of the upper part (smaller cone) to the frustum is:
\[ \boxed{1:7} \]