If the height of a cylinder is increased by 15 per cent and the radius of its base is decreased by 10 per cent then by what precent will its curved surface area change ?

Options

A. 3.5 per cent decrease
B. 3.5 per cent increase
C. 5 per cent increase
D. 5 per cent decrease

Answer

To solve the problem, we need to find the change in the curved surface area of a cylinder when the height is increased by 15% and the radius is decreased by 10%.

### Step-by-Step Solution:

1. **Understanding the Curved Surface Area (CSA) of a Cylinder**:
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Calculating the New Height**:
If the height is increased by 15%, the new height \( h' \) can be calculated as:
\[ h' = h + 0.15h = 1.15h \]

3. **Calculating the New Radius**:
If the radius is decreased by 10%, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.10r = 0.90r \]

4. **Finding the New Curved Surface Area**:
Now, substituting the new height and radius into the CSA formula:
\[ \text{New CSA} = 2\pi r' h' = 2\pi (0.90r)(1.15h) \]
Simplifying this:
\[ \text{New CSA} = 2\pi (0.90 \times 1.15) rh = 2\pi (1.035) rh \]

5. **Calculating the Percentage Change in CSA**:
The original CSA is:
\[ \text{Original CSA} = 2\pi rh \]
The percentage change in CSA can be calculated using the formula:
\[ \text{Percentage Change} = \left( \frac{\text{New CSA} - \text{Original CSA}}{\text{Original CSA}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left( \frac{2\pi (1.035) rh - 2\pi rh}{2\pi rh} \right) \times 100 \]
Simplifying this:
\[ \text{Percentage Change} = \left( \frac{2\pi rh (1.035 - 1)}{2\pi rh} \right) \times 100 = (0.035) \times 100 = 3.5\% \]

6. **Conclusion**:
The curved surface area of the cylinder increases by **3.5%**.