By what per cent does the volume of cylinder decrease , if there is 100% decrease in its radius and height ?

Answer

To solve the problem of how much the volume of a cylinder decreases when there is a 100% decrease in both its radius and height, we can follow these steps:

### 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 Calculation**:
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. **Calculate the New Dimensions**:
A 100% decrease in radius means the new radius \( r' \) is:
\[ r' = r - 100\% \text{ of } r = r - r = 0 \]
Similarly, a 100% decrease in height means the new height \( h' \) is:
\[ h' = h - 100\% \text{ of } h = h - h = 0 \]

4. **New Volume Calculation**:
The new volume \( V_2 \) with the new dimensions is:
\[ V_2 = \pi (r')^2 (h') = \pi (0)^2 (0) = 0 \]

5. **Volume Decrease Calculation**:
The decrease in volume can be calculated as:
\[ \text{Decrease in Volume} = V_1 - V_2 = \pi r^2 h - 0 = \pi r^2 h \]

6. **Percentage Decrease Calculation**:
The percentage decrease in volume is given by:
\[ \text{Percentage Decrease} = \left( \frac{\text{Decrease in Volume}}{V_1} \right) \times 100 = \left( \frac{\pi r^2 h}{\pi r^2 h} \right) \times 100 = 100\% \]

### Final Answer:
The volume of the cylinder decreases by **100%**.