If the radius of a sphere decreases by 24%, by what percentage does its total surface area decrease?
Aptitude
Volume and Surface Area
Difficulty: Easy
Choose an option
-
A44%
-
B49%
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C42.24%
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D46.2%
Answer
Correct Answer: 42.24%
Explanation
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:
S′/S = 0.76^2 = 0.5776Percent decrease = (1 − 0.5776) * 100% = 42.24%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%