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Thread: exponential growth, doubling

  1. #1
    syv
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    Question exponential growth, doubling

    Hello I'm a new user on this forum and I've difficulties with this question in my handbook regarding exponential growth.
    I'll post the question below and how i would solve it(seems wrong according to my handbook)

    In 1990, the chinese population was 3 times as high as the population of europe.
    The chinese population rises each year with 2% ,while the population of europe goes down with 0.5%.
    When will the chinese population be 4 times as high as the population of europe.

    I'd say: 400 * 1.02t = 100 * 0.995t
    400/100 = 0.995t/1.02t
    log (4) = t*log (0.995) - t*log(1.02)
    log (4) = t*(log (0.995) - log(1.02))
    log (4)/(log 0.995 - log 1.02) = t = -55.864

    This seems to be wrong. According to my handbook the answer is after 11.6 years.

    The answer in my handbook is a lot more logical than mine, and I really want to know what I did wrong.

    Thanks in advance!
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  2. #2
    MHF Contributor
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    Re: exponential growth, doubling

    In 1990, $C_0=3E_0$

    In year 1990+t (where t is a number of years):

    $E_t = 0.995^tE_0$
    $C_t = 1.02^tC_0$

    You want to know when will $C_t = 4E_t$. This is the same as saying $\dfrac{C_t}{E_t} = 4$

    $C_t = 1.02^tC_0 = 1.02^t\cdot 3E_0 = 3\left(\dfrac{1.02}{0.995}\right)^t E_t$

    Dividing both sides by $E_t$ gives:

    $\dfrac{C_t}{E_t} = 3\left( \dfrac{1.02}{0.995} \right)^t$

    Setting equal to 4 and solving for $t$ gives:

    $t = \dfrac{\ln\left(\dfrac{4}{3}\right)}{\ln\left( \dfrac{1.02}{0.995} \right)} \approx 11.5930$
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  3. #3
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    Re: exponential growth, doubling

    3*(1.02)^n = 4*(.995)^n
    n = 11.593 ...per Sir Slip
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  4. #4
    syv
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    Re: exponential growth, doubling

    Thanks for the fast reply!
    I would never come up with the first answer, but i understand it.

    But i do not get why DenisB says "3*(1.02)^n= 4*(0.995)^n"
    Now it seems, the population of europe will be 3/4th of the chinese instead of 1/4th. (my thought process might be pretty stupid, i'm sorry)
    I'd say "1*(1.02)^n = 4*(0.955)^n
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  5. #5
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    Re: exponential growth, doubling

    There's a "1" in there:

    3*(1.02)^n = 4*1*(.995)^n

    Hokay?!
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  6. #6
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    Re: exponential growth, doubling

    Does not matter what we start with, as long as it's 3:1
    Can be looked at as 2 savings acoounts, one
    earning -1/2% (like a Canadian bank!) and
    one earnimg 2%:
    Code:
    YR        EUROPE             CHINA
    0        1000.00           3000.00
    1   5.00  995.00    60.00  3060.00
    2   4.97  990.03    61.20  3121.20
    ...
    11  4.76  946.36    73.14  3730.12
    11.593 ?
    12  4.73  941.62    74.60  3804.72
    Got it?
    Last edited by DenisB; Jul 26th 2017 at 07:52 AM.
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