If the height of a cylinder is increased by 30%, and the radius of its base is decreased by 15%, then by what percentage will its curved surface area change?

Options

A. `10%`
B. `12%`
C. `8%`
D. `10.5%`

Answer

To solve the problem step by step, we need to find the change in the curved surface area of a cylinder when its height is increased by 30% and its radius is decreased by 15%.

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

### Step 2: Define initial dimensions
Let's assume the initial height \( h \) and radius \( r \) of the cylinder:
- Let the initial height \( h = 10 \) units (this is arbitrary for calculation).
- Let the initial radius \( r = 20 \) units (this is also arbitrary for calculation).

### Step 3: Calculate the new height after the increase
The height of the cylinder is increased by 30%.
- New height \( h' = h + 0.30h = 1.30h \)
- Substituting the initial height:
\[ h' = 1.30 \times 10 = 13 \text{ units} \]

### Step 4: Calculate the new radius after the decrease
The radius of the cylinder is decreased by 15%.
- New radius \( r' = r - 0.15r = 0.85r \)
- Substituting the initial radius:
\[ r' = 0.85 \times 20 = 17 \text{ units} \]

### Step 5: Calculate the initial curved surface area
Using the initial dimensions:
\[ \text{CSA}_{\text{initial}} = 2 \pi r h = 2 \pi (20)(10) = 400 \pi \]

### Step 6: Calculate the new curved surface area
Using the new dimensions:
\[ \text{CSA}_{\text{new}} = 2 \pi r' h' = 2 \pi (17)(13) = 442 \pi \]

### Step 7: Calculate the change in curved surface area
To find the change in curved surface area:
\[ \Delta \text{CSA} = \text{CSA}_{\text{new}} - \text{CSA}_{\text{initial}} = 442 \pi - 400 \pi = 42 \pi \]

### Step 8: Calculate the percentage change in curved surface area
The percentage change is given by:
\[ \text{Percentage Change} = \left( \frac{\Delta \text{CSA}}{\text{CSA}_{\text{initial}}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \left( \frac{42 \pi}{400 \pi} \right) \times 100 = \left( \frac{42}{400} \right) \times 100 = 10.5\% \]

### Final Answer
The curved surface area of the cylinder increases by **10.5%**.

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