If the radius of the cylinder is decreased by 20%, then by how much percent the height must be increased, so that the volume of the cylinder remains same?

Options

A. 44
B. `36.25`
C. `56.25`
D. `62.5`

Answer

To solve the problem, we need to determine how much the height of a cylinder must be increased when the radius is decreased by 20%, in order to keep the volume of the cylinder the same.

### 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.

2. **Initial Values**:
Let's assume the initial radius \( r \) is 100 cm and the initial height \( h \) is 1 cm.

3. **Calculate the New Radius**:
If the radius is decreased by 20%, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.2r = 100 - 20 = 80 \text{ cm} \]

4. **Set Up the Volume Equation**:
The initial volume \( V \) using the original radius and height is:
\[ V = \pi (100)^2 (1) = 10000\pi \]
The new volume using the new radius and the new height \( h' \) is:
\[ V' = \pi (80)^2 (h') = 6400\pi h' \]

5. **Equate the Volumes**:
To keep the volume the same, we set the two volumes equal:
\[ 10000\pi = 6400\pi h' \]

6. **Cancel \(\pi\)**:
Dividing both sides by \(\pi\):
\[ 10000 = 6400 h' \]

7. **Solve for \( h' \)**:
Rearranging gives:
\[ h' = \frac{10000}{6400} = \frac{1000}{640} = \frac{25}{16} \text{ cm} \]

8. **Calculate the Increase in Height**:
The original height \( h \) was 1 cm. The new height \( h' \) is \( \frac{25}{16} \) cm. The increase in height is:
\[ \text{Increase} = h' - h = \frac{25}{16} - 1 = \frac{25}{16} - \frac{16}{16} = \frac{9}{16} \text{ cm} \]

9. **Calculate the Percentage Increase**:
To find the percentage increase in height, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Height}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{\frac{9}{16}}{1} \right) \times 100 = \frac{9 \times 100}{16} = \frac{900}{16} = 56.25\% \]

### Final Answer:
The height must be increased by **56.25%** to keep the volume of the cylinder the same.