If the height of a given cone be doubled and radius of the base remains the same, the ratio of the volume of the given cone of that of the second cone will be

Options

A. `2:1`
B. `1:8`
C. `1:2`
D. `8:1`

Answer

To solve the problem, we need to find the ratio of the volumes of two cones: the original cone and a second cone with double the height but the same radius.

### Step-by-Step Solution:

1. **Understand the Volume Formula for a Cone**:
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cone.

2. **Define the Dimensions of the Original Cone**:
Let's assume the radius \( r \) of the original cone is \( r \) and the height \( h \) is \( h \). Therefore, the volume \( V_1 \) of the original cone is:
\[ V_1 = \frac{1}{3} \pi r^2 h \]

3. **Define the Dimensions of the Second Cone**:
According to the problem, the height of the second cone is doubled, so the new height \( h' \) is:
\[ h' = 2h \]
The radius remains the same, so \( r' = r \).

4. **Calculate the Volume of the Second Cone**:
The volume \( V_2 \) of the second cone can be calculated using the same volume formula:
\[ V_2 = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi r^2 (2h) = \frac{1}{3} \pi r^2 \cdot 2h = \frac{2}{3} \pi r^2 h \]

5. **Find the Ratio of the Volumes**:
Now, we need to find the ratio of the volume of the original cone \( V_1 \) to the volume of the second cone \( V_2 \):
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r^2 h}{\frac{2}{3} \pi r^2 h} \]
Simplifying this ratio:
\[ = \frac{1}{3} \cdot \frac{3}{2} = \frac{1}{2} \]

6. **Final Result**:
Therefore, the ratio of the volume of the original cone to that of the second cone is:
\[ \text{Ratio} = 1 : 2 \]