If the height of a given cone becomes four times and the radius of the base becomes twice, what is ratio of the volume of the given cone and the volume of the new cone ?

Options

A. `1:4`
B. `1:8`
C. `1:16`
D. `1:32`

Answer

To find the ratio of the volume of the given cone to the volume of the new cone after the height and radius have changed, we can follow these steps:

### Step 1: Write 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.

### Step 2: Calculate the volume of the given cone.
Let the original radius be \( r \) and the original height be \( h \). Therefore, the volume of the given cone \( V \) can be expressed as:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 3: Determine the new dimensions of the cone.
According to the problem:
- The height of the cone becomes four times, so the new height \( h' = 4h \).
- The radius of the base becomes twice, so the new radius \( r' = 2r \).

### Step 4: Calculate the volume of the new cone.
Now, we can calculate the volume of the new cone \( V' \):
\[ V' = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi (2r)^2 (4h) \]
Calculating \( (2r)^2 \):
\[ (2r)^2 = 4r^2 \]
Thus, substituting this back into the volume formula:
\[ V' = \frac{1}{3} \pi (4r^2)(4h) = \frac{1}{3} \pi (16r^2h) = \frac{16}{3} \pi r^2 h \]

### Step 5: Find the ratio of the volumes.
Now, we need to find the ratio of the volume of the given cone \( V \) to the volume of the new cone \( V' \):
\[ \text{Ratio} = \frac{V}{V'} = \frac{\frac{1}{3} \pi r^2 h}{\frac{16}{3} \pi r^2 h} \]
The \( \frac{1}{3} \pi r^2 h \) terms cancel out:
\[ \text{Ratio} = \frac{1}{16} \]

### Step 6: Express the ratio in the required format.
Thus, the ratio of the volume of the given cone to the volume of the new cone is:
\[ V : V' = 1 : 16 \]

### Conclusion
The final answer is:
\[ \text{The ratio of the volume of the given cone to the volume of the new cone is } 1 : 16. \]