For a right circular cylinder, the radius is increased by 20% and the height is increased by 15%. By what percentage does the curved surface area of the cylinder increase?

Introduction / Context:
This question tests percentage change concepts applied to geometry. The curved surface area of a cylinder depends on both its radius and height. Here, both radius and height are increased by given percentages, and we are asked to find the resulting overall percentage increase in curved surface area. This requires understanding of how multiplicative changes combine and avoiding the mistake of simply adding percentage increases directly.

Given Data / Assumptions:

• Initial radius of the cylinder = r.
• Initial height of the cylinder = h.
• Radius increases by 20%, so new radius = 1.20r.
• Height increases by 15%, so new height = 1.15h.
• We are concerned with the curved surface area (CSA).

Concept / Approach:
The curved surface area of a cylinder is given by:
CSA = 2 * pi * r * h If r and h change to r′ and h′, then new CSA is:
CSA′ = 2 * pi * r′ * h′ The ratio CSA′ / CSA equals (r′ / r) * (h′ / h). After finding this ratio, we convert it into percentage increase by subtracting 1 and multiplying by 100%.

Step-by-Step Solution:
New radius r′ = 1.20r. New height h′ = 1.15h. Original CSA = 2 * pi * r * h. New CSA = 2 * pi * (1.20r) * (1.15h). CSA′ = 2 * pi * r * h * (1.20 * 1.15). Compute factor: 1.20 * 1.15 = 1.38. Thus, CSA′ / CSA = 1.38. Percentage increase = (1.38 − 1) * 100% = 0.38 * 100% = 38%.

Verification / Alternative check:
We can check with simple values: let r = 10 and h = 10. Then original CSA = 2 * pi * 10 * 10 = 200pi. New r = 12, new h = 11.5, so new CSA = 2 * pi * 12 * 11.5 = 276pi. The ratio 276 / 200 = 1.38, again giving 38% increase. This numerical check confirms our percentage calculation.

Why Other Options Are Wrong:
33% or 35% would result from averaging or incorrectly adding the individual percentage changes instead of multiplying the scale factors. 41% and 44% are too high and would imply a larger combined factor than 1.38. The correct method uses multiplication of (1 + 0.20) and (1 + 0.15), not addition of 20% and 15% alone.

Common Pitfalls:
A frequent mistake is to simply add 20% and 15% to get 35%, ignoring the fact that the changes are multiplicative. Some students also mistakenly apply the formula for total surface area instead of curved surface area, which could change the result. It is crucial to work with ratios and scale factors when dealing with percentage changes in products like r * h.

Final Answer:
The curved surface area of the cylinder increases by 38%.