The height of a cone is 12 cm. and radius of its base is 3 cm. The cone is cut into two partts by a plane parallel to its base such same. What is the ratio of volume of upper part and lower part respectively ?

Options

A. `1:3`
B. `1:7`
C. `1:8`
D. `1:4`

Answer

To find the ratio of the volume of the upper part and the lower part of a cone that is cut by a plane parallel to its base, we can follow these steps:

### Step 1: Understand the dimensions of the cone
The height of the cone (H) is given as 12 cm, and the radius of its base (R) is given as 3 cm.

### Step 2: Calculate the volume of the original cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V = \frac{1}{3} \pi (3)^2 (12) = \frac{1}{3} \pi (9)(12) = \frac{1}{3} \pi (108) = 36\pi \text{ cm}^3 \]

### Step 3: Determine the height of the upper and lower parts
Since the cone is cut into two equal parts by a plane parallel to its base, the height of each part will be:
\[ \text{Height of upper part} = \text{Height of lower part} = \frac{12}{2} = 6 \text{ cm} \]

### Step 4: Calculate the radius of the upper part
The radius of the upper part of the cone will be proportional to its height. Since the original cone has a height of 12 cm and a radius of 3 cm, the radius of the upper part (r) can be found using similar triangles:
\[ \frac{r}{3} = \frac{6}{12} \implies r = 3 \times \frac{6}{12} = 1.5 \text{ cm} \]

### Step 5: Calculate the volume of the upper part
Now, we can calculate the volume of the upper part using the same volume formula:
\[ V_{\text{upper}} = \frac{1}{3} \pi (1.5)^2 (6) = \frac{1}{3} \pi (2.25)(6) = \frac{1}{3} \pi (13.5) = 4.5\pi \text{ cm}^3 \]

### Step 6: Calculate the volume of the lower part
The volume of the lower part can be found by subtracting the volume of the upper part from the volume of the original cone:
\[ V_{\text{lower}} = V - V_{\text{upper}} = 36\pi - 4.5\pi = 31.5\pi \text{ cm}^3 \]

### Step 7: Find the ratio of the volumes
The ratio of the volume of the upper part to the volume of the lower part is:
\[ \text{Ratio} = \frac{V_{\text{upper}}}{V_{\text{lower}}} = \frac{4.5\pi}{31.5\pi} = \frac{4.5}{31.5} = \frac{1}{7} \]

### Final Answer
The ratio of the volume of the upper part to the volume of the lower part is \( 1 : 7 \).

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