If the radius of a sphere decreases by 24%, by what percentage does its total surface area decrease?

Introduction / Context:Area scales as the square of a linear dimension. A decrease in radius reduces surface area by the square of the scale factor.

Given Data / Assumptions:

• Initial radius r; decreased radius r′ = 0.76 r (since 24% decrease).
• Surface area S = 4πr^2 and S′ = 4πr′^2.

Concept / Approach:Compute ratio S′/S = (r′/r)^2 = (0.76)^2 and convert to percent decrease.

Step-by-Step Solution:

Verification / Alternative check:Square 76/100 precisely to avoid rounding errors; 76^2 = 5776; move decimal: 0.5776.

Why Other Options Are Wrong:

• 44%, 49%, 46.2%: Not equal to the exact square-based reduction.

Common Pitfalls:Subtracting 24% twice or assuming area decreases by exactly double (48%).

Final Answer:42.24%