If the radius of a cylinder is decreased by 50% and the height is increased by 50% to form a new cylinder, the volume will be decreased by

Options

A. `0%`
B. `25%`
C. `62.5%`
D. `75%`

Answer

To solve the problem, we need to determine how the volume of a cylinder changes when the radius is decreased by 50% and the height is increased by 50%.

### Step-by-Step Solution:

1. **Define the Original Cylinder Dimensions:**
- Let the original radius of the cylinder be \( r \).
- Let the original height of the cylinder be \( h \).

2. **Calculate the Original Volume:**
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

3. **Determine the New Dimensions:**
- The radius is decreased by 50%, so the new radius \( r' \) is:
\[ r' = r - 0.5r = 0.5r \]
- The height is increased by 50%, so the new height \( h' \) is:
\[ h' = h + 0.5h = 1.5h \]

4. **Calculate the New Volume:**
- The new volume \( V' \) of the cylinder with the new dimensions is:
\[ V' = \pi (r')^2 h' = \pi (0.5r)^2 (1.5h) \]
- Simplifying this gives:
\[ V' = \pi (0.25r^2)(1.5h) = \pi (0.375r^2h) \]

5. **Compare the Volumes:**
- The original volume was \( V = \pi r^2 h \).
- The new volume is \( V' = \pi (0.375r^2h) \).
- To find the decrease in volume, we can express the new volume as a fraction of the original volume:
\[ \frac{V'}{V} = \frac{0.375r^2h}{r^2h} = 0.375 \]

6. **Calculate the Decrease in Volume:**
- The decrease in volume can be calculated as:
\[ \text{Decrease} = V - V' = V - 0.375V = (1 - 0.375)V = 0.625V \]
- This means the volume decreases by 62.5% of the original volume.

### Conclusion:
The volume of the new cylinder is decreased by **62.5%**.