If height of cylinder is decreased by 8%, while its radius remains unchanged, by what per cent does the volume decrease?

Options

A. 0.04
B. 0.06
C. 0.08
D. 0.1

Answer

To solve the problem of how much the volume of a cylinder decreases when its height is decreased by 8% while its radius remains unchanged, we can follow these steps:

### 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 initial volume**:
Let the initial height of the cylinder be \( h \). Therefore, the initial volume \( V \) is:
\[ V = \pi r^2 h \]

3. **Calculate the new height after the decrease**:
The height is decreased by 8%, which means the new height \( h' \) can be calculated as:
\[ h' = h - 0.08h = 0.92h \]

4. **Calculate the new volume**:
The new volume \( V' \) after the height reduction is:
\[ V' = \pi r^2 h' = \pi r^2 (0.92h) = 0.92 \pi r^2 h \]

5. **Determine the decrease in volume**:
The decrease in volume \( \Delta V \) can be calculated as:
\[ \Delta V = V - V' = \pi r^2 h - 0.92 \pi r^2 h = (1 - 0.92) \pi r^2 h = 0.08 \pi r^2 h \]

6. **Calculate the percentage decrease in volume**:
The percentage decrease in volume is given by:
\[ \text{Percentage Decrease} = \left( \frac{\Delta V}{V} \right) \times 100 = \left( \frac{0.08 \pi r^2 h}{\pi r^2 h} \right) \times 100 \]
Simplifying this gives:
\[ \text{Percentage Decrease} = 0.08 \times 100 = 8\% \]

### Final Answer:
The volume of the cylinder decreases by **8%**.