In a cylinder, if radius of the base is halved and height is twiced, then the curved surface area will be _______ the curved surface area of the original cylinder.

Options

A. Two times
B. Three times
C. Same as
D. Four times

Answer

To solve the problem, we need to compare the curved surface area of the original cylinder with the curved surface area of the modified cylinder where the radius is halved and the height is doubled.

### Step-by-Step Solution:

1. **Understand the formula for the curved surface area of a cylinder**:
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi R H \]
where \( R \) is the radius and \( H \) is the height of the cylinder.

2. **Define the original dimensions**:
Let the original radius of the cylinder be \( R \) and the original height be \( H \).

3. **Calculate the original curved surface area**:
Using the formula:
\[ \text{Original CSA} = 2\pi R H \]

4. **Define the new dimensions**:
According to the problem:
- The radius is halved:
\[ \text{New Radius} = \frac{R}{2} \]
- The height is doubled:
\[ \text{New Height} = 2H \]

5. **Calculate the new curved surface area**:
Substitute the new radius and height into the CSA formula:
\[ \text{New CSA} = 2\pi \left(\frac{R}{2}\right)(2H) \]
Simplifying this:
\[ \text{New CSA} = 2\pi \cdot \frac{R}{2} \cdot 2H = 2\pi RH \]

6. **Compare the original and new curved surface areas**:
- Original CSA: \( 2\pi R H \)
- New CSA: \( 2\pi R H \)

We see that:
\[ \text{New CSA} = \text{Original CSA} \]

### Conclusion:
The curved surface area of the modified cylinder is the same as the curved surface area of the original cylinder. Therefore, the answer is that the curved surface area will be **the same** as the curved surface area of the original cylinder.