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# City Growth

## Question

A town's population of size x doubled after 30 years (2x).  How long ago was this population 1/2x?

## 5 answers to this question

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Spoiler

If the population growth is susceptible to be modelled as exponential growth with a constant rate of growth then the answer is 30 years ago, as the doubling time is constant.

Or as an example would show it visually.

 Growth Rate 2.33739% X 100.0000 X 100.0000 X+1 102.3374 X-1 97.7160 X+2 104.7294 X-2 95.4842 X+3 107.1773 X-3 93.3033 X+4 109.6825 X-4 91.1722 X+5 112.2462 X-5 89.0899 X+6 114.8698 X-6 87.0551 X+7 117.5548 X-7 85.0667 X+8 120.3025 X-8 83.1238 X+9 123.1144 X-9 81.2252 X+10 125.9921 X-10 79.3700 X+11 128.9370 X-11 77.5572 X+12 131.9508 X-12 75.7858 X+13 135.0350 X-13 74.0549 X+14 138.1913 X-14 72.3635 X+15 141.4214 X-15 70.7107 X+16 144.7269 X-16 69.0956 X+17 148.1098 X-17 67.5175 X+18 151.5717 X-18 65.9754 X+19 155.1145 X-19 64.4685 X+20 158.7401 X-20 62.9960 X+21 162.4505 X-21 61.5572 X+22 166.2476 X-22 60.1512 X+23 170.1334 X-23 58.7774 X+24 174.1101 X-24 57.4349 X+25 178.1798 X-25 56.1231 X+26 182.3445 X-26 54.8412 X+27 186.6066 X-27 53.5887 X+28 190.9683 X-28 52.3647 X+29 195.4320 X-29 51.1687 X+30 200.0000 X-30 50.0000

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Spoiler

If we can infer (OP does not say so, but ...) the town's population doubles every 30 years,
then it had population (1/2) x 30 years before it had population x.

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Sixty years.  Perhaps splitting hairs but I'm a word guy.  The OP states that the town's population has already "doubled" (past tense).

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1 hour ago, plainglazed said:
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Sixty years.  Perhaps splitting hairs but I'm a word guy.  The OP states that the town's population has already "doubled" (past tense).

I intended your interpretation but came up with a much different answer than the ones reported

On 9/9/2017 at 1:55 PM, araver said:
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If the population growth is susceptible to be modelled as exponential growth with a constant rate of growth then the answer is 30 years ago, as the doubling time is constant.

Or as an example would show it visually.

 Growth Rate 2.33739% X 100.0000 X 100.0000 X+1 102.3374 X-1 97.7160 X+2 104.7294 X-2 95.4842 X+3 107.1773 X-3 93.3033 X+4 109.6825 X-4 91.1722 X+5 112.2462 X-5 89.0899 X+6 114.8698 X-6 87.0551 X+7 117.5548 X-7 85.0667 X+8 120.3025 X-8 83.1238 X+9 123.1144 X-9 81.2252 X+10 125.9921 X-10 79.3700 X+11 128.9370 X-11 77.5572 X+12 131.9508 X-12 75.7858 X+13 135.0350 X-13 74.0549 X+14 138.1913 X-14 72.3635 X+15 141.4214 X-15 70.7107 X+16 144.7269 X-16 69.0956 X+17 148.1098 X-17 67.5175 X+18 151.5717 X-18 65.9754 X+19 155.1145 X-19 64.4685 X+20 158.7401 X-20 62.9960 X+21 162.4505 X-21 61.5572 X+22 166.2476 X-22 60.1512 X+23 170.1334 X-23 58.7774 X+24 174.1101 X-24 57.4349 X+25 178.1798 X-25 56.1231 X+26 182.3445 X-26 54.8412 X+27 186.6066 X-27 53.5887 X+28 190.9683 X-28 52.3647 X+29 195.4320 X-29 51.1687 X+30 200.0000 X-30 50.0000

forgive my lack of parenthesis what I meant for the problem is 1/(2x)

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I think I got what the answer is, now that we know it is 1/(2x).

Spoiler

Since x is in the denominator, the answer will depend on what x is. This means the answer I got was an equation.

If y is the amount of years ago, then the answer is: y=(-30 log(2x^2))/log(2)

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