A plane parallel to the base of a cone cuts its axis in the ratio 2:1 from its base. The ratio of the volume of the lower part of the cone to that of the upper part is

Options

A. A)`27:1`
B. B)`26:1`
C. C)`8:1`
D. D)`7:1`

Answer

To solve the problem, we need to find the ratio of the volume of the lower part of the cone to that of the upper part after a plane cuts the cone's axis in the ratio 2:1 from its base.

### Step-by-Step Solution:

1. **Understanding the Cone and the Cut**:
- Let's denote the height of the entire cone as \( h \).
- According to the problem, the cone's axis is divided into two parts by a plane: the lower part (from the base to the cut) is \( 2x \) and the upper part (from the cut to the apex) is \( x \). Thus, the total height of the cone is \( h = 2x + x = 3x \).

2. **Finding the Volume of the Entire Cone**:
- The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
- For the entire cone, substituting \( h = 3x \):
\[ V_{\text{total}} = \frac{1}{3} \pi r^2 (3x) = \pi r^2 x \]

3. **Finding the Volume of the Upper Cone**:
- The height of the upper cone (the smaller cone) is \( x \).
- The radius of the upper cone can be found using similar triangles. Since the height of the upper cone is \( x \) (1 part), and the total height is \( 3x \) (3 parts), the radius of the upper cone is:
\[ r_{\text{upper}} = \frac{1}{3} r \]
- Now, we can find the volume of the upper cone:
\[ V_{\text{upper}} = \frac{1}{3} \pi \left(\frac{1}{3} r\right)^2 x = \frac{1}{3} \pi \left(\frac{1}{9} r^2\right) x = \frac{1}{27} \pi r^2 x \]

4. **Finding the Volume of the Lower Part**:
- The volume of the lower part of the cone is the volume of the entire cone minus the volume of the upper cone:
\[ V_{\text{lower}} = V_{\text{total}} - V_{\text{upper}} = \pi r^2 x - \frac{1}{27} \pi r^2 x \]
- To combine these volumes, we find a common denominator:
\[ V_{\text{lower}} = \frac{27}{27} \pi r^2 x - \frac{1}{27} \pi r^2 x = \frac{26}{27} \pi r^2 x \]

5. **Finding the Ratio of Volumes**:
- Now we have:
- Volume of the lower part: \( V_{\text{lower}} = \frac{26}{27} \pi r^2 x \)
- Volume of the upper part: \( V_{\text{upper}} = \frac{1}{27} \pi r^2 x \)
- The ratio of the volume of the lower part to the upper part is:
\[ \text{Ratio} = \frac{V_{\text{lower}}}{V_{\text{upper}}} = \frac{\frac{26}{27} \pi r^2 x}{\frac{1}{27} \pi r^2 x} = \frac{26}{1} = 26:1 \]

### Final Answer:
The ratio of the volume of the lower part of the cone to that of the upper part is \( 26:1 \).