Results 1 to 11 of 11

Thread: How to calculate rate of future population decline with loss and gain parameters?

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    Canada
    Posts
    5

    How to calculate rate of future population decline with loss and gain parameters?

    I need help figuring out what I think is an algebra problem, but I don't know how to set it up.
    I have a population starting at 1.6 billion. The annual growth rate is 2%. The annual mortality rate is 0.6%.

    The problem is this: In addition to the above parameters, there is a constant loss of 52,560,000 people per year.

    How many years would it take for the population to drop to 620,000,000?

    (also could you please show how you did it?)

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,216
    Thanks
    3705

    Re: How to calculate rate of future population decline with loss and gain parameters?

    population (in billions) as a function of time in years ...

    $P(t) = 1.6(1.014)^t - 0.05256t$

    $0.62 = 1.6(1.014)^t - 0.05256t$

    $t \approx 45.5$ years (solved with a calculator)

    note that after about 62 years, the population will begin increasing again ...
    Last edited by skeeter; Nov 20th 2015 at 11:17 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,255
    Thanks
    500

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Hmmm...I get not quite 27 years.

    160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
    n = 26.968614....

    Population at zero at end of 40th year.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,216
    Thanks
    3705

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Quote Originally Posted by DenisB View Post
    Hmmm...I get not quite 27 years.

    160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
    n = 26.968614....

    Population at zero at end of 40th year.
    there is a constant loss of 52,560,000 people per year.
    ?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,255
    Thanks
    500

    Re: How to calculate rate of future population decline with loss and gain parameters?

    That's where we differ.
    I'm assuming this "loss" reduces the effect of the 1.4% growth.
    Like, I'm looking at it as "taking it to the Bank"!
    A savings account earning 1.4% annually,
    from which an annual withdrawal is made:
    Code:
    YEAR   WITHDRAWAL   INTEREST    BALANCE
      0                             160,000
      1     -5,256       2,240      156,984
     .....
     26     -5,256         986       66,193
     27     -5,256         927       61,863
    That's how I see it...probably wrong !
    But I'll stick to my water pistols
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Sep 2013
    From
    USA
    Posts
    632
    Thanks
    194

    Re: How to calculate rate of future population decline with loss and gain parameters?

    1.6*exp(0.014*t) - 0.52*t

    Growth/decay are exponential with natural base e. Banks use discontinuous interest except in the case of very large companies with continuous cash flow.

    How to calculate rate of future population decline with loss and gain parameters?-untitled2.gif
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jan 2013
    From
    Canada
    Posts
    5

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Hey guys, I appreciate you tackling this problem! Let me give you a bit of background on where this is coming from, so you understand what I'm trying to accomplish.

    I'm an author. I'm creating a science fiction universe and I have to have internal consistency while determining the rate of decline of the inhabitants of my extremely overpopulated nation-state as they go into the future with some very unethical measures of drastically culling the population. They exterminate a set number of individuals per year, with an aim to get down to a particular population number. I want to know how long it would take them, but I can't just take the current population, subtract the desired population, then divide that number by the constant number eliminated per year, because that doesn't take into account population gain and loss parameters of birth and mortality rates. Since these gains and losses are based on percentages, as the population changes each year by the constant extermination number, the resulting numbers of gains and losses must change dynamically as they reflect that updated total each and every year. Therefore, unless I can determine all three of these factors together, I can't get an accurate estimate on how long it will take my population to be culled to its desired size.

    And of course the reason I am asking if you can show me how you did it is because I might have to do it again in the future, if aspects of the plot shift which affect the population of the state. (for example, to begin with, the nation occupied the whole continent of Australia = 8.5 million km^2, then I decided to devote 40% of that continent to wastelands, so the area dropped to 6.4 million km^2. To keep the population density proportionate, the entire starting population also has to drop by 40%. Therefore, I have to do all of these prospective population calculations again. And I'd like to know an equation I can just plug the numbers into, in the likely case that I have to shift things again.

    I hope all that casts a little more light on the convoluted situation? Hahaha
    Thanks! I really do appreciate the help. As is the case with many authors, math is not my strong suit. ;-)
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,216
    Thanks
    3705

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Quote Originally Posted by DenisB View Post
    Hmmm...I get not quite 27 years.

    160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
    n = 26.968614....

    Population at zero at end of 40th year.
    did it again using the DE, $\dfrac{dP}{dt} = 0.014P - .05256$

    my solution agrees (real close) to Denis ... t about 26.78 years
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,255
    Thanks
    500

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Quote Originally Posted by votan View Post
    1.6*exp(0.014*t) - 0.52*t
    True nuff, Mr. Votan.
    BUT how precise do we need to be with a population approaching 2 billion
    and applying growth/decline factors that are guessed at to start with?

    Geezzz...you'd need to end up with people not quite dead, plus a few
    housewives pregnant for 4.123456789.... months

    As far as the "spitit of the problem goes", I see no problem
    in rounding out (even to the closest 1000) ALL calculations.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Jan 2013
    From
    Canada
    Posts
    5

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Quote Originally Posted by skeeter View Post
    did it again using the DE, $\dfrac{dP}{dt} = 0.014P - .05256$

    my solution agrees (real close) to Denis ... t about 26.78 years

    Not sure the meaning of the dP/dt, equation, but it is agreed that this works?

    160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
    n = 26.968614

    So the answer being approximately 27 years. Thanks!
    (Does the "n-1" in this case have anything to do with degrees of freedom, or is that another, unrelated n-1? )
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    2,255
    Thanks
    500

    Re: How to calculate rate of future population decline with loss and gain parameters?

    Quote Originally Posted by PistolSlap View Post
    160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
    n = 26.968614

    So the answer being approximately 27 years. Thanks!
    (Does the "n-1" in this case have anything to do with degrees of freedom, or is that another, unrelated n-1? )
    The -1 has nothing to do with the n;
    after 1.014^n is calculated, 1 is deducted.
    Example: 1.02^7 - 1 = 1.1486 - 1 = .1486
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Population Decline
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Apr 16th 2014, 07:12 PM
  2. gain/loss ratios?
    Posted in the Business Math Forum
    Replies: 0
    Last Post: Dec 13th 2010, 05:11 PM
  3. Calculate Elevation Gain and Loss
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Oct 13th 2010, 05:54 PM
  4. Capital Gain or Loss of selling a bond
    Posted in the Business Math Forum
    Replies: 3
    Last Post: Jul 9th 2010, 04:13 PM
  5. Calculate the gain in kinetic energy of each electron.
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: May 8th 2008, 06:25 PM

Search Tags


/mathhelpforum @mathhelpforum