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

Options

A. 1 : 2
B. 1 : 4
C. 1 : 6
D. 1 : 8

Answer

To find the ratio of the volumes of the upper part of a cone and the entire cone when the cone is cut by a horizontal plane passing through the midpoint of its axis, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Geometry of the Cone**:
- Let the height of the cone be \( h \).
- When the cone is cut horizontally at the midpoint, the height of the upper part (small cone) is \( \frac{h}{2} \) and the height of the lower part (remaining cone) is also \( \frac{h}{2} \).

2. **Identify the Radii**:
- Let the radius of the base of the full cone be \( R \).
- Let the radius of the base of the smaller cone (upper part) be \( r \).

3. **Use Similar Triangles**:
- The triangles formed by the height and radius of the cones are similar. Thus, we can set up a ratio using the properties of similar triangles.
- The triangles are:
- Triangle ABC (full cone)
- Triangle ADE (smaller cone)
- Since the triangles are similar, we have:
\[ \frac{r}{R} = \frac{\frac{h}{2}}{h} \]
- Simplifying this gives:
\[ \frac{r}{R} = \frac{1}{2} \]
- Therefore, we can express \( R \) in terms of \( r \):
\[ R = 2r \]

4. **Calculate the Volume of the Upper Cone**:
- The volume \( V_1 \) of the smaller cone (upper part) is given by the formula:
\[ V_1 = \frac{1}{3} \pi r^2 \left(\frac{h}{2}\right) = \frac{1}{6} \pi r^2 h \]

5. **Calculate the Volume of the Full Cone**:
- The volume \( V_2 \) of the full cone is:
\[ V_2 = \frac{1}{3} \pi R^2 h \]
- Substituting \( R = 2r \):
\[ V_2 = \frac{1}{3} \pi (2r)^2 h = \frac{1}{3} \pi (4r^2) h = \frac{4}{3} \pi r^2 h \]

6. **Find the Ratio of the Volumes**:
- Now, we need to find the ratio of the volume of the upper part to the volume of the full cone:
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{\frac{1}{6} \pi r^2 h}{\frac{4}{3} \pi r^2 h} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{1/6}{4/3} = \frac{1}{6} \times \frac{3}{4} = \frac{3}{24} = \frac{1}{8} \]

7. **Final Answer**:
- Therefore, the ratio of the volumes of the upper part to the full cone is:
\[ \text{Ratio} = 1 : 8 \]