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A certain bacteria colony doubles in size every day for 20 [#permalink]

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12 Dec 2012, 13:42

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A certain bacteria colony doubles in size every day for 20 days, at which point it reaches the limit of its habitat and can no longer grow. If two bacteria colonies start growing simultaneously, how many days will it take them to reach the habitat’s limit?

Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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12 Dec 2012, 15:09

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Let the size at the colony at the start of 20 days be = 1 At the end of the first day it would be 2...At the end of the second day it would be 2(2) and the day after that (2)(2)(2)...You can see then that following the same pattern at the end of 20 days you have (2)^20 since the population doubles... Hope this helps...

A certain bacteria colony doubles in size every day for 20 days, at which point it reaches the limit of its habitat and can no longer grow. If two bacteria colonies start growing simultaneously, how many days will it take them to reach the habitat’s limit? 6.33 7.5 10 15 19

Say the colony has x bacteria in the beginning. 1st day - 2x bacteria 2nd day - \(2^2x\) bacteria 3rd day - \(2^3x\) bacteria . 20th day - \(2^{20}x\) bacteria

If we start with 2 bacteria colonies, it means there are already 2x bacteria (equal to number of bacteria at the end of the first day in the first case). In how many more days will they reach \(2^{20}x\) ? Another 19 days!
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A certain bacteria colony doubles in size every day for 20 days, at which point it reaches the limit of its habitat and can no longer grow. If two bacteria colonies start growing simultaneously, how many days will it take them to reach the habitat’s limit?

A. 6.33 B. 7.5 C. 10 D. 15 E. 19

If there is one bacteria colony, then it will reach the limit of its habitat in 20 days.

If there are two bacteria colonies, then in order to reach the limit of habitat they would need to double one time less than in case with one colony. Thus colonies need to double 19 times.

Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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15 Dec 2012, 06:34

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I am not sure whether ny nethod works for everybody but this is how i did.

So let say our limit is 5 days (just for easiness of calculation) and we have one collony with 1 bacterium in the first day, so in 5 days we have 1+2+4+8+16=31 bacteria, but if we have two colonies, then in order to fill the max capacity of 31 bacteria we need overall 4 days (1+2+4+8=15 and 1+2+4+8=15).

The same principle applies for 20 days, then we need 19 days to fill the max capacity with two colonies. E is the answer.

Hope thats helpful!
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Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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16 Dec 2012, 04:21

Ans: If there is one bacteria colony, then it will reach the limit of its habitat in 20 days which is 2^19

If there are two bacteria colonies growing simultaneously, then they would multiply as 2, 2^2, 2^3 in order to reach the limit of habitat they would need one day less than normal which is 19 days. (E).
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Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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20 Jan 2014, 06:27

Let the final amount of bacteria or the MAX size of the habitat on day 20 be 2x. With single colony doubling every day, the quantity of bacteria on day 19 will be 2x/2 which is x. Now we have another colony growing at the same pace, its size on day 19 is also going to be x. Therefore on day 19 colony 1 size is x and colony 2 size is also x. So total size on day 19 will be 2x. Which is the MAX size of the habitat. Hence E is the answer.

PS: The question seems a little ambiguous because we have to assume each bacteria colony starts with same amount in the beginning.
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A certain bacteria colony doubles in size every day for 20 [#permalink]

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08 Apr 2016, 20:25

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Attached is a visual that should help. Technically the equation for "doubles in size x times" should be \(n(2^x)\) because we do not know that the colony is necessarily doubling in integer form, i.e, 1,2,4,8 etc. But it's easier to eliminate that part since "n" would be on both sides of the equation, and we are not given a specific starting number so we might as well use the integer powers of two.

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Screen Shot 2016-04-08 at 8.24.37 PM.png [ 72.01 KiB | Viewed 3458 times ]

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Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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15 Oct 2016, 12:56

I found this formula to be easy to apply.

Final population growth = S * P ^ (t/l) S = starting population P = progression (doubles = 2, triples = 3 etc.) t/l = total amount of iterations t = time I = intervals

Let x be final population or habitat limit = > x = 1 bacteria * 2 ^(20 days/1 day) which is x = 2 ^20 if we have 2 bacterias, then each has to fill in half of the habitat => x/2 = 2^20 / 2 = > x/2 = 2 ^19 - so it will take 19 days for each bacteria to fill in half of the habitat, which mens that together they will reach habitat's limit in 19 days

Re: A certain bacteria colony doubles in size every day for 20 [#permalink]

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23 Oct 2017, 12:09

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