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Exponential Growth- Am I doing this right?

  1. Feb 12, 2013 #1
    1. Let's say that at the end of 2010, the population of Athens was 100,000. Let's also say that the relative growth rate of the population was 2% per year.
    What's the population at t = 0.5 years after the end of 2010?

    2. To solve this sort of problem, I thought I'd have to use: y= A(1+b)t

    A is the initial population

    b is the growth factor

    t is the time


    I thought that since the population growth rate was 2% per year, the growth factor would have to be (.001 + 1).

    I figured out the .001 by saying: (.2%/yr) = (x/.5yr)
    x= .1% or .001
    (.1%/100%) = .001 = b

    y= 100,000(1 + .001).5

    y= (100,000)(1.001)

    y= 100,100 people

    Is this right?
    Thanks for the help! :)
     
  2. jcsd
  3. Feb 12, 2013 #2

    Mark44

    Staff: Mentor

    You're mixing up percentages and decimal numbers. .2% is a small fraction of 1%.
     
  4. Feb 12, 2013 #3

    haruspex

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    As a check, try calculating the population after 1 year. Does the result seem reasonable in relation to your answer for 6 months? (Hint: it isn't.)
    But there's a subtler problem. Over such a timescale, population growth is approximately a continuous process. That is, in a small time δt, the population will increase by a factor 1+λδt. That gives you a differential equation. Solve that, then use the annual growth rate to determine λ.
     
  5. Feb 12, 2013 #4
    What a minute. Wouldn't my whole (.2%/1yr)=(x /100%) be unnecessary in the first place?

    Wouldn't I need to do (2%/100%) = .02

    Shouldn't it instead be: y= 100,000(1 + .02).5 ???

    y= (100,000)(1.00995)

    y= 100,995.0
     
  6. Feb 12, 2013 #5

    haruspex

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    Yes, that's fine.
     
  7. Feb 12, 2013 #6
    Oh, okay. Thank you! :D
     
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