The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its :
curved surface area

Answer

To find the percentage change in the curved surface area of a solid right circular cylinder when the radius decreases by 20% and the height increases by 10%, follow these steps:

### Step 1: Define the initial parameters
Let the initial radius of the cylinder be \( r \) and the initial height be \( h \).

### Step 2: Calculate the initial curved surface area
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ A = 2 \pi r h \]

### Step 3: Determine the new radius
The radius decreases by 20%. Therefore, the new radius \( r' \) can be calculated as:
\[ r' = r - 0.2r = 0.8r \]

### Step 4: Determine the new height
The height increases by 10%. Therefore, the new height \( h' \) can be calculated as:
\[ h' = h + 0.1h = 1.1h \]

### Step 5: Calculate the new curved surface area
Now, substitute \( r' \) and \( h' \) into the formula for the curved surface area:
\[ A' = 2 \pi r' h' = 2 \pi (0.8r)(1.1h) \]
\[ A' = 2 \pi (0.88rh) = 1.76 \pi rh \]

### Step 6: Calculate the percentage change in area
Now, we need to find the percentage change in the curved surface area:
\[ \text{Percentage Change} = \frac{A' - A}{A} \times 100 \]
Substituting the values of \( A' \) and \( A \):
\[ \text{Percentage Change} = \frac{1.76 \pi rh - 2 \pi rh}{2 \pi rh} \times 100 \]
\[ = \frac{(1.76 - 2) \pi rh}{2 \pi rh} \times 100 \]
\[ = \frac{-0.24 \pi rh}{2 \pi rh} \times 100 \]
\[ = \frac{-0.24}{2} \times 100 = -12\% \]

### Conclusion
The percentage change in the curved surface area is a decrease of **12%**.

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