Percent decrease in sphere surface area when radius shrinks: If the radius of a sphere is decreased by 24%, by what percent does its surface area decrease?

Introduction / Context:Sphere surface area S = 4πr^2, so S scales with r^2. A percentage change in r results in a squared factor for S.

Given Data / Assumptions:r_new = 0.76 r (since 24% decrease).

Concept / Approach:Compute S_new/S_old = (0.76)^2 and convert to percent decrease.

Step-by-Step Solution:S_new/S_old = 0.76^2 = 0.5776Percent decrease = (1 − 0.5776)*100% = 42.24%

Verification / Alternative check:Small changes approximate as 2Δr, but exact method is squaring the factor.

Why Other Options Are Wrong:49% and 44% are rough guesses; 46.2% corresponds to compounding a different percentage.

Common Pitfalls:Applying linear decrease to area; forgetting the square dependence.

Final Answer:42.24%