In a cylinder, if radius is doubled and height is halved, curved surface area will be halved (b) doubled (c) same (d) four times

Options

A. halved
B. doubled
C. same
D. four time

Explanation

To solve the problem, we need to analyze how the curved surface area of a cylinder changes when the radius is doubled and the height is halved.

### 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. **Modify the dimensions according to the problem**:
According to the problem, the radius is doubled and the height is halved. Therefore:
- New radius = \( 2r \)
- New height = \( \frac{h}{2} \)

5. **Calculate the new curved surface area**:
Substitute the new dimensions into the CSA formula:
\[ \text{New CSA} = 2 \pi (2r) \left(\frac{h}{2}\right) \]

6. **Simplify the new curved surface area**:
Now simplify the expression:
\[ \text{New CSA} = 2 \pi (2r) \left(\frac{h}{2}\right) = 2 \pi \cdot 2r \cdot \frac{h}{2} \]
The \( 2 \) in the numerator and the \( 2 \) in the denominator cancel out:
\[ \text{New CSA} = 2 \pi r h \]

7. **Compare the original and new curved surface areas**:
From our calculations:
- Original CSA = \( 2 \pi r h \)
- New CSA = \( 2 \pi r h \)

Thus, the new curved surface area is the same as the original curved surface area.

8. **Conclusion**:
Therefore, the curved surface area remains the same when the radius is doubled and the height is halved. The correct answer is (c) same.