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

### Concept & Successive Percentage Change The curved surface area ($CSA$) of a cylinder is directly proportional to the product of its radius and height ($CSA = 2\pi rh$). We can use the successive percentage change formula: $$\text{Net Change \%} = a + b + \frac{ab}{100}$$ where $a$ and $b$ are the percentage changes (positive for increase, negative for decrease). ### Step-by-Step Solution 1. **Identify the changes:** - Height increases by $15\% \Rightarrow a = +15$ - Radius decreases by $10\% \Rightarrow b = -10$ 2. **Apply the successive percentage formula:** - $\text{Net Change \%} = 15 + (-10) + \frac{15 \times (-10)}{100}$ - $\text{Net Change \%} = 5 - \frac{150}{100}$ - $\text{Net Change \%} = 5 - 1.5 = +3.5$ 3. **Interpret the result:** - Since the result is positive, it represents an increase. - The curved surface area increases by $3.5\%$. ### Exam Strategy & Shortcut Using the effective percentage formula ($a + b + \frac{ab}{100}$) is much faster than assuming values like $r=100$, $h=100$ and calculating the actual area before and after. ### Common Pitfall A common error is just adding the percentages ($+15 - 10 = +5\%$) and ignoring the multiplicative effect ($\frac{ab}{100}$). Always use the full formula for products. ### Final Answer Therefore, the correct answer is **$3.5 \text{ percent increase}$**.