Multiple choice

The diameters of three circles are in the ratio $3 : 5 : 6$ . If the sum of the circumferences of these circles be $308: cm$; find the difference between the areas of the largest and the smallest of these circles.

  1. $1039.5 \: cm^{2}$
  2. $1339.5 \: cm^{2}$
  3. $1239.5 \: cm^{2}$
  4. $1139.5 \: cm^{2}$
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A Correct answer
Explanation

Let diameters be 3k, 5k, 6k. Circumferences are pi*3k + pi*5k + pi*6k = 14*pi*k = 308. So k = 308/(14*pi) = 22/pi. Radii are 1.5k, 2.5k, 3k. Difference in areas = pi*(3k)^2 - pi*(1.5k)^2 = pi*k^2*(9 - 2.25) = 6.75*pi*k^2. Substituting k = 22/pi gives 6.75*pi*(22/pi)^2 = 6.75 * 484 / (22/7) = 1039.5.

AI explanation

Using the diameter ratio of 3:5:6, let the diameters be 3x, 5x, and 6x, making their circumferences 3x, 5x, and 6x times pi. Setting the sum of the circumferences to 308 gives 14x times 22/7 equals 308, so x equals 7, meaning the largest and smallest diameters are 42 cm and 21 cm respectively. Their radii are 21 cm and 10.5 cm, making the difference in their areas pi times (21 squared minus 10.5 squared), which equals 22/7 times 330.75. This calculates to 1039.5 square cm.

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