If in a cylinder , both height and radius increase by 100% , by what per cent does its volume increase ?

Answer

To solve the problem of how much the volume of a cylinder increases when both its height and radius are increased by 100%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for 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. **Initial Dimensions**:
Let the initial radius be \( r \) and the initial height be \( h \).

3. **Calculate Initial Volume**:
The initial volume \( V_1 \) can be calculated as:
\[ V_1 = \pi r^2 h \]

4. **Increase Dimensions by 100%**:
If both the radius and height are increased by 100%, the new radius \( r' \) and new height \( h' \) will be:
\[ r' = r + 100\% \text{ of } r = r + r = 2r \]
\[ h' = h + 100\% \text{ of } h = h + h = 2h \]

5. **Calculate New Volume**:
The new volume \( V_2 \) can be calculated as:
\[ V_2 = \pi (r')^2 (h') = \pi (2r)^2 (2h) = \pi (4r^2)(2h) = 8\pi r^2 h \]

6. **Calculate the Increase in Volume**:
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V_2 - V_1 = 8\pi r^2 h - \pi r^2 h = (8 - 1)\pi r^2 h = 7\pi r^2 h \]

7. **Calculate the Percentage Increase in Volume**:
The percentage increase in volume can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V_1} \right) \times 100\% \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{7\pi r^2 h}{\pi r^2 h} \right) \times 100\% = 7 \times 100\% = 700\% \]

### Final Answer:
The volume of the cylinder increases by **700%**.