If the radius of a cylinder is increased by `25%` , by how much per cent the height must be reduced , so that the volume of the cylinder remains the same .

Options

A. 36
B. 56
C. 64
D. 46

Answer

To solve the problem, we need to find out how much the height of the cylinder must be reduced when the radius is increased by 25% in order to keep the volume of the cylinder constant.

### 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. **Increase in Radius**:
If the radius is increased by 25%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.25r = 1.25r \]

4. **New Volume with Increased Radius**:
The new volume \( V_2 \) with the increased radius and the new height \( h' \) is:
\[ V_2 = \pi (r')^2 h' = \pi (1.25r)^2 h' = \pi (1.5625r^2) h' \]

5. **Setting Volumes Equal**:
Since we want the volume to remain constant, we set \( V_1 \) equal to \( V_2 \):
\[ \pi r^2 h = \pi (1.5625r^2) h' \]

6. **Canceling Out Common Terms**:
We can cancel \( \pi \) and \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = 1.5625 h' \]

7. **Solving for New Height**:
Rearranging gives us:
\[ h' = \frac{h}{1.5625} \]

8. **Calculating the Reduction in Height**:
To find out how much the height must be reduced, we calculate:
\[ \text{Reduction} = h - h' = h - \frac{h}{1.5625} = h \left(1 - \frac{1}{1.5625}\right) \]

9. **Finding the Fraction**:
Calculate \( \frac{1}{1.5625} \):
\[ \frac{1}{1.5625} = 0.64 \]
Therefore:
\[ \text{Reduction} = h(1 - 0.64) = h(0.36) \]

10. **Calculating Percentage Reduction**:
The percentage reduction in height is:
\[ \text{Percentage Reduction} = \frac{\text{Reduction}}{h} \times 100 = 0.36 \times 100 = 36\% \]

### Final Answer:
The height must be reduced by **36%**.