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A runner starts running on a circular path of radius 'r'

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A runner starts running on a circular path of radius 'r' [#permalink] New post 18 Jun 2013, 21:16
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Difficulty:

  65% (hard)

Question Stats:

58% (01:56) correct 42% (03:05) wrong based on 12 sessions
A runner starts running on a circular path of radius 'r' meters. His average speed (in meters/min) is ∏r during the first 30 sec, ∏r/2 during the next one minute, ∏r/4 during the next 2 min, ∏r/8 during the next 4 min and so on... What is the ratio of the nth round to the previous round?

A) 4
B) 8
C) 16
D) 32
E) 64
[Reveal] Spoiler: OA

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Re: A runner starts running on a circular path of radius 'r' [#permalink] New post 19 Jun 2013, 21:20
atalpanditgmat wrote:
A runner starts running on a circular path of radius 'r' meters. His average speed (in meters/min) is ∏r during the first 30 sec, ∏r/2 during the next one minute, ∏r/4 during the next 2 min, ∏r/8 during the next 4 min and so on... What is the ratio of the nth round to the previous round?


I believe the last sentence should say: What is the ratio of the time taken for the nth round to that for the previous round? Without stating time, the question can't really be answered.

Given that information, though:

Time = Distance / Velocity

In the first interval: Velocity = (∏r meters/min) => (∏r/2 meters in 30 seconds) => Quarter-Lap in 30 seconds: T = 30 seconds, D = Quarter-Lap, Velocity = ∏r

Since the velocity is being halved and the time is being doubled, the distance is clearly constant: D = V * T = (V_old * 0.5) * (T_old * 2) = D_old

So at each velocity and time shift, we are covering a quarter-round of distance. We need to compare the times to complete 4-interval rounds.

Since T = D/V, and D is constant, T will vary inversely with V.

Consider how the velocity for a quarter round in one cycle will change with respect to a quarter round in the next cycle:

Since the velocity is halved each time, it will be sixteenth-ed after each series of 4 quarter-rounds. (1/2 * 1/2 * 1/2 * 1/2 = 1/16).

So since each round's velocity is 1/16 of the previous round's velocity, the time is multiplied by 16.

So, -C-.
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Re: A runner starts running on a circular path of radius 'r' [#permalink] New post 20 Jun 2013, 00:10
atalpanditgmat wrote:
A runner starts running on a circular path of radius 'r' meters. His average speed (in meters/min) is ∏r during the first 30 sec, ∏r/2 during the next one minute, ∏r/4 during the next 2 min, ∏r/8 during the next 4 min and so on... What is the ratio of the nth round to the previous round?

A) 4
B) 8
C) 16
D) 32
E) 64



The problem is faulty. The sequence (∏r during the first 30 sec, ∏r/2 during the next one minute, ∏r/4 during the next 2 min, ∏r/8 during the next 4 min and so on...) is an infinite GP series. If the person keeps on taking double the time for next half distance, he can never reach the end point to complete 1 round.

Distance series , ∏r + ∏r/2+ ∏r/4+ ∏r/8+ ∏r/16+ ∏r/32+ .... +∏r/2^(n-1) = 2∏r

∏r(1- 0.5^n)/(1-0.5) = 2∏r
1-0.5^n = 1
0.5^n = 0
n = infinite

So the time series for round 1: 30 + 60 + ... infinite times is undefined.

what is the source of this problem ?

He will keep on reducing the speed and finally in an infinite time he will complete 1 circle.
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Re: A runner starts running on a circular path of radius 'r' [#permalink] New post 20 Jun 2013, 08:03
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ConnectTheDots wrote:

Distance series , ∏r + ∏r/2+ ∏r/4+ ∏r/8+ ∏r/16+ ∏r/32+ .... +∏r/2^(n-1) = 2∏r


This is where your error is - look at the velocity and time again, and you'll find that these are not the distances travelled.
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Re: A runner starts running on a circular path of radius 'r'   [#permalink] 20 Jun 2013, 08:03
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A runner starts running on a circular path of radius 'r'

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