If the height of cylinder is increased by 35% and radius is increased by 10%, then what will be the percentage increase in curved surface area of cylinder?

Options

A. `46.5`
B. 45
C. `48.5`
D. `49.7`

Answer

To solve the problem, we need to find the percentage increase in the curved surface area of a cylinder when its height and radius are increased by certain percentages.

### Step 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:
\[ \text{CSA} = 2\pi rh \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Define the original dimensions.
Let the original radius be \( r \) and the original height be \( h \).

### Step 3: Calculate the new dimensions after the increase.
- The new radius after a 10% increase:
\[ r' = r + 0.10r = 1.10r \]
- The new height after a 35% increase:
\[ h' = h + 0.35h = 1.35h \]

### Step 4: Calculate the new curved surface area.
Using the new dimensions, the new curved surface area (CSA') is:
\[ \text{CSA}' = 2\pi r'h' = 2\pi (1.10r)(1.35h) \]
\[ \text{CSA}' = 2\pi (1.10)(1.35)(rh) = 2\pi (1.485)(rh) \]

### Step 5: Calculate the percentage increase in curved surface area.
The original curved surface area (CSA) is:
\[ \text{CSA} = 2\pi rh \]
The percentage increase in curved surface area is given by:
\[ \text{Percentage Increase} = \left( \frac{\text{CSA}' - \text{CSA}}{\text{CSA}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{2\pi (1.485)(rh) - 2\pi (rh)}{2\pi (rh)} \right) \times 100 \]
\[ = \left( \frac{1.485 - 1}{1} \right) \times 100 = 0.485 \times 100 = 48.5\% \]

### Final Answer:
The percentage increase in the curved surface area of the cylinder is **48.5%**.

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