Effect on cylinder volume when radius increases: If the cylinder’s radius increases by 25% while height stays the same, what is the percentage increase in volume?

Introduction / Context:
Volume depends on r^2h. With height fixed, the percent change in V is governed by the square of the radius scale factor. This tests understanding of quadratic dependence on r.

Given Data / Assumptions:

• r increases by 25% → r_new = 1.25r
• h unchanged
• V_new / V_old = (r_new^2 / r^2) = (1.25)^2 = 1.5625

Concept / Approach:
Compute the area factor from the squared radius ratio, then convert the factor to a percentage increase: (1.5625 − 1) × 100%.

Step-by-Step Solution:
Scale factor in r = 1.25Scale factor in r^2 = 1.25^2 = 1.5625Percent increase = 0.5625 × 100% = 56.25%

Verification / Alternative check:
Pick r = 4, h = 10: V_old = π * 16 * 10 = 160π. New r = 5 → V_new = π * 25 * 10 = 250π. Increase = 90π → 90/160 = 56.25% increase.

Why Other Options Are Wrong:
52.25% and 50.4% are not equal to (1.25^2 − 1); 60.26% is an overestimate; 25% confuses linear with quadratic dependence.

Common Pitfalls:
Applying 25% directly to volume; forgetting the square on r in πr^2h.

Final Answer:
56.25%