If the height of a cylinder becomes `1/4` of the original height and the radius is doubled, then which of the following will be true?

Options

A. Curved surface area of the cylinder will be doubled.
B. Curved surface area of the cylinder will remain unchanged.
C. Curved surface area of the cylinder will be halved.
D. Curved surface area will be `1/4` of the original curved surface.

Answer

To solve the problem, we need to analyze how the changes in height and radius affect the curved surface area of the cylinder.

### Step-by-Step Solution:

1. **Understand the Original Dimensions:**
- Let the original height of the cylinder be \( h \).
- Let the original radius of the cylinder be \( r \).

2. **Calculate the Original Curved Surface Area:**
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
- Substituting the original dimensions:
\[ \text{Original CSA} = 2 \pi r h \]

3. **Determine the New Dimensions:**
- The new height of the cylinder becomes \( \frac{1}{4}h \) (one-fourth of the original height).
- The new radius of the cylinder becomes \( 2r \) (double the original radius).

4. **Calculate the New Curved Surface Area:**
- Using the new dimensions in the CSA formula:
\[ \text{New CSA} = 2 \pi (2r) \left(\frac{1}{4}h\right) \]
- Simplifying this:
\[ \text{New CSA} = 2 \pi (2r) \cdot \frac{1}{4}h = 2 \pi \cdot 2r \cdot \frac{1}{4}h = \frac{2 \pi r h}{2} = \pi r h \]

5. **Compare the Original and New Curved Surface Areas:**
- The original curved surface area was \( 2 \pi r h \).
- The new curved surface area is \( \pi r h \).
- Therefore, the new curved surface area is half of the original curved surface area:
\[ \text{New CSA} = \frac{1}{2} \times \text{Original CSA} \]

### Conclusion:
The curved surface area of the cylinder will be half of the original curved surface area.