A circle's radius increases by 1%. By what percentage does its area increase?
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Answer: D. 2.01%
Explanation
The area of a circle scales proportionally with the square of its radius according to geometric formulas. Increasing the radius by one percent results in a binomial expansion yielding two percent plus the squared term. Calculating two times one percent plus zero point zero one percent gives a total area increase of two point zero one percent. Area scaling laws state that changing any linear dimension by a certain factor alters the area by that factor squared. Percentage approximations for small increments use the first two terms of binomial expansions for rapid estimation. Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter across all sizes.
Option A incorrectly assumes the area increases linearly at the exact same rate as the radius dimension. Option B suggests an excessively high percentage increase that ignores standard circle area scaling laws. Option C overestimates the squared contribution, miscalculating the binomial expansion result for small percentage increments.
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- Mathematics
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- Mathematics
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- Difficulty
- medium
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