Multiple choice

Three circles of radii $ \displaystyle r_{1},r_{2}$ and $r_{3}$ are drawn concentric to each other. The radii $ \displaystyle r_{1}$ and $r_{2}$ are such that the area of the circle with radius $ \displaystyle r_{1}$ is equal to the area between the circles of radius $ \displaystyle r_{2}$ and $r_{1}$. The area between the circles of radii $ \displaystyle r_{3}$ and $r_{2}$ is equal to area between the circles of radii $ \displaystyle r_{2}$ and $r_{r}$. What is the value of $ \displaystyle r_{1}:r_{2}:r_{3}?$

  1. $ \displaystyle 1:\sqrt{2}:\sqrt{3}$
  2. $1:2:3$
  3. $1:\sqrt2:3$
  4. $1:2:\sqrt3$
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A Correct answer
Explanation

The first condition gives r2^2 - r1^2 = r1^2, so r2^2 = 2r1^2. The second condition gives r3^2 - r2^2 = r1^2, so r3^2 = 3r1^2. Therefore, the ratio is 1:sqrt(2):sqrt(3).

AI explanation

The area of the circle with radius r1 is pi times r1 squared, and making it equal to the area between r2 and r1 gives pi times r1 squared equals pi times r2 squared minus pi times r1 squared. Simplifying this relationship yields 2 times r1 squared equals r2 squared, which means r1 divided by r2 is 1 divided by the square root of 2. Setting the area between r2 and r1 equal to the area between r3 and r2 gives pi times r1 squared equals pi times r3 squared minus pi times r2 squared, and substituting r2 squared as 2 times r1 squared results in pi times r3 squared equals 3 times pi times r1 squared. Thus, r1 divided by r3 is 1 divided by the square root of 3, making the overall ratio r1 to r2 to r3 equal to 1 to the square root of 2 to the square root of 3.

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