The radius of the base and height of a right circular cylinder are each increased by `20%`. The volume of the cylinder will be increased by :

Options

A. `40%`
B. `60%`
C. `72.8%`
D. `96%`

Answer

To solve the problem of how much the volume of a right circular cylinder increases when both the radius and height are increased by 20%, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Original Dimensions**:
Let the original radius of the base of the cylinder be \( R \) and the original height be \( H \).

2. **Calculate the New Dimensions**:
- The new radius after a 20% increase:
\[ R' = R + 0.2R = 1.2R \]
- The new height after a 20% increase:
\[ H' = H + 0.2H = 1.2H \]

3. **Calculate the Original Volume**:
The volume \( V_1 \) of the original cylinder is given by the formula:
\[ V_1 = \pi R^2 H \]

4. **Calculate the New Volume**:
The new volume \( V_2 \) of the cylinder with the new dimensions is:
\[ V_2 = \pi (R')^2 (H') = \pi (1.2R)^2 (1.2H) \]
Simplifying this:
\[ V_2 = \pi (1.44R^2)(1.2H) = 1.728 \pi R^2 H \]

5. **Calculate the Increase in Volume**:
The increase in volume is:
\[ \text{Increase} = V_2 - V_1 = 1.728 \pi R^2 H - \pi R^2 H = (1.728 - 1) \pi R^2 H = 0.728 \pi R^2 H \]

6. **Calculate the Percentage Increase in Volume**:
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{V_1} \right) \times 100 = \left( \frac{0.728 \pi R^2 H}{\pi R^2 H} \right) \times 100 \]
Simplifying this:
\[ \text{Percentage Increase} = 0.728 \times 100 = 72.8\% \]

### Final Answer:
The volume of the cylinder will be increased by **72.8%**.