If the height and radii of the cylinder halved then the volume of the cylinder becomes ____

Options

A. 8 times the original volume
B. 4 times the original volume
C. `1/8` time the original volume
D. it will rename same

Answer

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

### Step-by-Step Solution:

1. **Understand the formula for the volume of a 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 of the cylinder.

2. **Define the original dimensions**:
Let the original radius be \( R \) and the original height be \( H \). Therefore, the original volume \( V_{original} \) can be expressed as:
\[ V_{original} = \pi R^2 H \]

3. **Halve the dimensions**:
If we halve the radius and height, the new radius \( r \) and new height \( h \) will be:
\[ r = \frac{R}{2} \quad \text{and} \quad h = \frac{H}{2} \]

4. **Calculate the new volume**:
Substitute the new radius and height into the volume formula:
\[ V_{new} = \pi \left(\frac{R}{2}\right)^2 \left(\frac{H}{2}\right) \]

5. **Simplify the expression**:
Calculate \( \left(\frac{R}{2}\right)^2 \):
\[ \left(\frac{R}{2}\right)^2 = \frac{R^2}{4} \]
Now substitute this back into the volume formula:
\[ V_{new} = \pi \left(\frac{R^2}{4}\right) \left(\frac{H}{2}\right) \]
This simplifies to:
\[ V_{new} = \pi \frac{R^2 H}{4 \cdot 2} = \pi \frac{R^2 H}{8} \]

6. **Relate the new volume to the original volume**:
We can see that:
\[ V_{new} = \frac{1}{8} \pi R^2 H = \frac{1}{8} V_{original} \]

### Conclusion:
Thus, when the height and radius of the cylinder are halved, the volume of the cylinder becomes:
\[ \frac{1}{8} \text{ of the original volume} \]