Multiple choice

The sum of the radii of two circles is $84\ cm$ and the difference of their areas is $5544$ $cm^2$. Calculate the radii of the two circles.

  1. $52.5\ cm,\ 31.5\ cm$
  2. $52.5\ cm,\ 30.5\ cm$
  3. $50.5\ cm,\ 31.5\ cm$
  4. $50.5\ cm,\ 30.5\ cm$
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A Correct answer
Explanation

r1 + r2 = 84. pi * (r1^2 - r2^2) = 5544. pi * (r1 - r2)(r1 + r2) = 5544. (22/7) * (r1 - r2) * 84 = 5544. (r1 - r2) * 264 = 5544. r1 - r2 = 21. Solving r1+r2=84 and r1-r2=21: 2*r1 = 105 => r1 = 52.5. r2 = 31.5.

AI explanation

Let the two unknown radii be R and r, with their sum given as 84 cm. The difference of their areas is given by the formula pi * R^2 - pi * r^2 = 5544, which factors to pi * (R + r) * (R - r) = 5544. Substituting the sum of 84 gives (22/7) * 84 * (R - r) = 5544, which simplifies to 264 * (R - r) = 5544. Solving this gives the difference between the radii as 21 cm. Using the sum of 84 and the difference of 21 in the elimination method yields the two radii of 52.5 cm and 31.5 cm.

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