If the height of a given cone becomes thrice and the radius of the bae remains the same, what is the ratio of the volume of the given cone and the volume of the second cone ?

Options

A. `1:3`
B. `1:9`
C. `1:sqrt(3)`
D. `1:27`

Answer

To solve the problem, we need to find the ratio of the volume of the original cone to the volume of the new cone after the height has been tripled while keeping the radius the same.

### Step-by-Step Solution:

1. **Understand the formula for the volume of 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 original cone's dimensions**:
Let the original height of the cone be \( h \) and the radius of the base be \( r \). Therefore, the volume \( V \) of the original cone can be expressed as:
\[ V = \frac{1}{3} \pi r^2 h \]

3. **Determine the dimensions of the new cone**:
According to the problem, the height of the cone becomes thrice its original height. Thus, the new height \( h' \) is:
\[ h' = 3h \]
The radius remains the same, so \( r' = r \).

4. **Calculate the volume of the new cone**:
Using the same volume formula for the new cone, we have:
\[ V' = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi r^2 (3h) \]
Simplifying this, we get:
\[ V' = \frac{1}{3} \pi r^2 \cdot 3h = \pi r^2 h \]

5. **Find the ratio of the volumes**:
Now, we need to find the ratio of the volume of the original cone \( V \) to the volume of the new cone \( V' \):
\[ \text{Ratio} = \frac{V}{V'} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h} \]
The \( \pi r^2 h \) terms cancel out, leaving us with:
\[ \text{Ratio} = \frac{1}{3} \]

6. **Express the ratio in standard form**:
Thus, the ratio of the volume of the original cone to the volume of the new cone is:
\[ 1 : 3 \]

### Final Answer:
The ratio of the volume of the given cone to the volume of the second cone is \( 1 : 3 \).