If the height of a cylinder becomes `(1)/(2)` of the original height and the radius is doubled, then volume of cylinder becomes ________ of its original volume.

Options

A. 2 times
B. `(1)/(2)` times
C. `(1)/(4)` times
D. 3 times

Answer

To solve the problem, we need to determine how the volume of a cylinder changes when its height is halved and its radius is doubled.

### Step-by-Step Solution:

1. **Identify the original volume of the cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

2. **Define the new dimensions**:
- The new height \( h' \) is half of the original height:
\[ h' = \frac{h}{2} \]
- The new radius \( r' \) is double the original radius:
\[ r' = 2r \]

3. **Calculate the new volume**:
The new volume \( V' \) of the cylinder with the new dimensions is:
\[ V' = \pi (r')^2 (h') \]
Substituting the values of \( r' \) and \( h' \):
\[ V' = \pi (2r)^2 \left(\frac{h}{2}\right) \]
Simplifying this:
\[ V' = \pi (4r^2) \left(\frac{h}{2}\right) = \pi \cdot 4r^2 \cdot \frac{h}{2} = 2\pi r^2 h \]

4. **Relate the new volume to the original volume**:
Now, we can express \( V' \) in terms of the original volume \( V \):
\[ V' = 2\pi r^2 h = 2V \]

5. **Conclusion**:
Therefore, the new volume \( V' \) is twice the original volume \( V \):
\[ V' = 2V \]

### Final Answer:
The volume of the cylinder becomes **2 times** its original volume.