If base diameter of a cylinder is increased by 50%, then by how much per cent its height must be decreased so as to keep its volume unaltered?

Options

A. `45.56%`
B. `55.56%`
C. `50.16%`
D. `62.33%`

Answer

To solve the problem, we need to determine how much the height of a cylinder must be decreased when the base diameter is increased by 50%, in order to keep the volume of the cylinder unchanged.

### 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.

2. **Determine the change in the radius**:
If the base diameter is increased by 50%, the new diameter \( D' \) can be expressed as:
\[ D' = D + 0.5D = 1.5D \]
Since the radius \( r \) is half of the diameter, the new radius \( r' \) will be:
\[ r' = \frac{D'}{2} = \frac{1.5D}{2} = 0.75D \]
Thus, the radius is increased by 50%.

3. **Express the new radius in terms of the original radius**:
Let the original radius be \( r \). Therefore, the new radius can be expressed as:
\[ r' = 1.5r \]

4. **Set up the equation for volume conservation**:
Since the volume must remain unchanged, we can set up the equation:
\[ \pi (r')^2 h' = \pi r^2 h \]
Substituting \( r' = 1.5r \):
\[ \pi (1.5r)^2 h' = \pi r^2 h \]
Simplifying this gives:
\[ 2.25r^2 h' = r^2 h \]

5. **Solve for the new height \( h' \)**:
Dividing both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ 2.25h' = h \]
Therefore, the new height \( h' \) is:
\[ h' = \frac{h}{2.25} = \frac{h}{\frac{9}{4}} = \frac{4h}{9} \]

6. **Calculate the percentage decrease in height**:
The original height is \( h \) and the new height is \( \frac{4h}{9} \). The decrease in height is:
\[ \text{Decrease} = h - \frac{4h}{9} = \frac{9h}{9} - \frac{4h}{9} = \frac{5h}{9} \]
To find the percentage decrease:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{Original Height}}\right) \times 100 = \left(\frac{\frac{5h}{9}}{h}\right) \times 100 = \frac{5}{9} \times 100 \approx 55.56\% \]

### Final Answer:
The height must be decreased by approximately **55.56%** to keep the volume of the cylinder unaltered.