If the radius of a cylinder is increased by 25% and its height remains unchanged, then find the per cent increase in volume.

Options

A. `56.25%`
B. `52.25%`
C. `50.4%`
D. `60.26 %`

Answer

To solve the problem, we need to find the percentage increase in the volume of a cylinder when its radius is increased by 25% while keeping the height constant.

### 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. **Let the original radius and height**:
Let the original radius of the cylinder be \( r \) and the height be \( h \).

3. **Calculate the original volume**:
The original volume \( V_1 \) of the cylinder is:
\[ V_1 = \pi r^2 h \]

4. **Increase the radius by 25%**:
If the radius is increased by 25%, the new radius \( r' \) will be:
\[ r' = r + 0.25r = 1.25r \]

5. **Calculate the new volume**:
The new volume \( V_2 \) with the increased radius is:
\[ V_2 = \pi (r')^2 h = \pi (1.25r)^2 h = \pi (1.5625r^2) h = 1.5625 \pi r^2 h \]

6. **Find the increase in volume**:
The increase in volume \( \Delta V \) is:
\[ \Delta V = V_2 - V_1 = 1.5625 \pi r^2 h - \pi r^2 h = (1.5625 - 1) \pi r^2 h = 0.5625 \pi r^2 h \]

7. **Calculate the percentage increase in volume**:
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left(\frac{\Delta V}{V_1}\right) \times 100 = \left(\frac{0.5625 \pi r^2 h}{\pi r^2 h}\right) \times 100 \]
Simplifying this gives:
\[ \text{Percentage Increase} = 0.5625 \times 100 = 56.25\% \]

### Final Answer:
The percentage increase in the volume of the cylinder is **56.25%**.