The radius of a cylinder is reduced to 50% of its actual radius. If its volume remains the same as before, then its height become k times the original height. The value of k is.

Options

A. 2
B. 16
C. 8
D. 4

Answer

To solve the problem, we need to analyze the relationship between the radius, height, and volume of a cylinder.

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
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 of the cylinder.

2. **Initial Volume**:
Let the initial radius be \( r \) and the initial height be \( h \). Therefore, the initial volume \( V_1 \) is:
\[ V_1 = \pi r^2 h \]

3. **New Radius**:
The radius is reduced to 50% of its original value. Thus, the new radius \( r' \) is:
\[ r' = \frac{r}{2} \]

4. **New Volume with Reduced Radius**:
The volume of the cylinder with the new radius \( r' \) and a new height \( h' \) is:
\[ V_2 = \pi (r')^2 h' = \pi \left(\frac{r}{2}\right)^2 h' = \pi \left(\frac{r^2}{4}\right) h' = \frac{\pi r^2 h'}{4} \]

5. **Volume Equality**:
Since the volume remains the same, we set \( V_1 \) equal to \( V_2 \):
\[ \pi r^2 h = \frac{\pi r^2 h'}{4} \]

6. **Canceling Common Terms**:
We can cancel \( \pi r^2 \) from both sides (assuming \( r \neq 0 \)):
\[ h = \frac{h'}{4} \]

7. **Finding New Height**:
Rearranging gives us:
\[ h' = 4h \]
This means the new height \( h' \) is 4 times the original height \( h \).

8. **Finding k**:
We define \( k \) as the factor by which the height has increased:
\[ k = \frac{h'}{h} = \frac{4h}{h} = 4 \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{4} \]