Cone with fixed height — radius increased by 15%: effect on volume: For a right circular cone, the base radius increases by 15% while the height remains fixed. By what percentage does the volume increase?

Introduction / Context:
With height fixed, a cone’s volume is proportional to r^2. A percentage increase in radius thus compounds as the square of the radius factor for the volume.

Given Data / Assumptions:

• r → 1.15 r.
• h unchanged.
• V ∝ r^2 when h is fixed.

Concept / Approach:
Compute the volume scale factor as (1.15)^2. Convert that factor into a percentage increase by subtracting 1 and multiplying by 100%.

Step-by-Step Solution:
New factor = (1.15)^2 = 1.3225Percent increase = (1.3225 − 1)*100% = 32.25%

Verification / Alternative check:
For small p, the approximation would be ≈ 2p = 30%, but the exact square gives 32.25%, slightly higher.

Why Other Options Are Wrong:
30% is a linear approximation; 31% and 34.75% are off the exact square; 32.25% matches the precise compounding.

Common Pitfalls:
Applying the linear percentage directly to volume rather than squaring the radius factor.

Final Answer:
32.25%