A cone is cut at mid point of its height by a frustum parallel to its base. The ratio between the two parts of cone would be

Options

A. `1:1`
B. `1:8`
C. `1:4`
D. `1:7`

Answer

To solve the problem of finding the ratio between the two parts of the cone when it is cut at the midpoint of its height by a frustum parallel to its base, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Cone and Frustum**:
- A cone is a three-dimensional geometric shape with a circular base and a single vertex (apex).
- When the cone is cut at the midpoint of its height, the upper part forms a smaller cone, and the lower part forms a frustum (the portion of the cone that remains).

2. **Identifying the Heights**:
- Let the total height of the cone be \( h \).
- Since the cone is cut at the midpoint, the height of the upper cone (smaller cone) is \( \frac{h}{2} \) and the height of the frustum (lower part) is also \( \frac{h}{2} \).

3. **Volume of the Cone**:
- The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
- Let \( R \) be the radius of the base of the original cone. The volume of the original cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi R^2 h \]

4. **Volume of the Smaller Cone**:
- The smaller cone that is formed after the cut has a height of \( \frac{h}{2} \) and a radius that is proportional to the height. Since the cones are similar, the radius of the smaller cone can be calculated as:
\[ r = \frac{R}{2} \]
- Therefore, the volume of the smaller cone is:
\[ V_{\text{small cone}} = \frac{1}{3} \pi \left(\frac{R}{2}\right)^2 \left(\frac{h}{2}\right) = \frac{1}{3} \pi \frac{R^2}{4} \frac{h}{2} = \frac{1}{24} \pi R^2 h \]

5. **Volume of the Frustum**:
- The volume of the frustum can be calculated by subtracting the volume of the smaller cone from the volume of the original cone:
\[ V_{\text{frustum}} = V_{\text{cone}} - V_{\text{small cone}} = \frac{1}{3} \pi R^2 h - \frac{1}{24} \pi R^2 h \]
- To perform this subtraction, we need a common denominator:
\[ V_{\text{frustum}} = \frac{8}{24} \pi R^2 h - \frac{1}{24} \pi R^2 h = \frac{7}{24} \pi R^2 h \]

6. **Finding the Ratio**:
- Now, we can find the ratio of the volumes of the smaller cone to the frustum:
\[ \text{Ratio} = \frac{V_{\text{small cone}}}{V_{\text{frustum}}} = \frac{\frac{1}{24} \pi R^2 h}{\frac{7}{24} \pi R^2 h} = \frac{1}{7} \]

### Final Answer:
The ratio between the two parts of the cone (the smaller cone and the frustum) is \( 1:7 \).