If the radius of a right circular cylinder open at both the ends, is decreased by 25% and the height of the cylinder is increased by 25%. Then the curved surface area of the cylinder thus formed

Options

A. remains unaltered
B. is increased by 25%
C. is increased by 6.25%
D. is decreased by 6.25%

Answer

To find the curved surface area of the modified cylinder after changes in its dimensions, we can follow these steps:

### Step-by-step Solution:

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

2. **Calculate the New Radius**:
- The radius is decreased by 25%. Therefore, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.25r = 0.75r \]

3. **Calculate the New Height**:
- The height is increased by 25%. Therefore, the new height \( h' \) can be calculated as:
\[ h' = h + 0.25h = 1.25h \]

4. **Formula for Curved Surface Area**:
- The formula for the curved surface area \( A \) of a right circular cylinder is given by:
\[ A = 2\pi r h \]

5. **Calculate the New Curved Surface Area**:
- Substitute the new radius \( r' \) and new height \( h' \) into the formula:
\[ A' = 2\pi (0.75r)(1.25h) \]
- Simplifying this expression:
\[ A' = 2\pi \times 0.75 \times 1.25 \times r \times h \]
\[ A' = 2\pi \times 0.9375 \times r \times h \]
\[ A' = 1.875\pi rh \]

6. **Conclusion**:
- The new curved surface area of the cylinder is \( 1.875\pi rh \).