Relative speed on circular and rectangular tracks

CAT 2005 Slot 1 · QA · Medium · Arithmetic

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A jogging park has two identical circular tracks touching each other, and a rectangular track enclosing the two circles. The edges of the rectangle are tangential to the circles. Two friends, A and B, start jogging simultaneously form the point where one of the circular tracks touches the smaller side of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they take the same time to return to their starting point?

  1. A.

    3.88%

  2. B.

    4.22%

  3. C.

    4.44%

  4. D.

    4.72%

Answer

D

Explanation

Let rr be the radius of each circular track. The rectangular track has dimensions 4r×2r4r \times 2r. Distance covered by A =2(4r+2r)=12r= 2(4r + 2r) = 12r. Distance covered by B =2×2πr=4πr= 2 \times 2\pi r = 4\pi r. For both to take equal time, ratio of speeds SBSA=4πr12r=π3\frac{S_B}{S_A} = \frac{4\pi r}{12r} = \frac{\pi}{3}. Percentage by which B is faster than A =(π−33)×100%≈3.1416−33×100%=4.72%= \left(\frac{\pi - 3}{3}\right) \times 100\% \approx \frac{3.1416 - 3}{3} \times 100\% = 4.72\%.

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