Results 1 to 6 of 6

Math Help - exponential growth

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    52

    exponential growth

    The question is if the population takes 20 years to triple, how many years does it take to double.

    \displaystyle y'=Ce^{k20}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member courteous's Avatar
    Joined
    Aug 2008
    From
    big slice of heaven
    Posts
    206
    We know that the population in 20 years ( y_{20}) is triple the starting one ( y_0):
    y_{20}=3y_0=y_0 e^{20k}

    It follows that
    e^{20k}=3
    and
    k=\frac{\ln(3)}{20}.

    Now set up equation for doubling in unknown  x time:
    y_x=y_0 e^{x\frac{\ln(3)}{20}}=2y_0

    and get from that

    e^{x\frac{\ln(3)}{20}}=2

    x\frac{\ln(3)}{20}}=\ln(2)

    x=20\frac{\ln(2)}{\ln(3)}=12.62
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    Is this the actual equation given to you?

    Does y' represent a derivative here?

    EDIT: Is k a constant or is it the time in years?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Feb 2010
    Posts
    52
    That is the actual question, and k is a constant. The equation was made by me from my interpretation.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    Okay, then a more appropriate equation would be:

    y = y_oe^{kt}

    Then you say, when t = 20 years, the population y is 3 times that of the initial population y_o

    3y_o = y_oe^{k(20)}

    This gives you e^{20k} = 3

    And then you continue like courteous did.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Exponentials to all bases and logarithms to all bases are interchangable. I prefer to avoid "e" in favor of an obvious base.

    Saying that a population take "T" years to triple is the same as saying it is given by P(t)= C3^{t/T} since "t/T" is the number of times it triples. So asking for the number of years to double is asking form t such that C3^{t/T}= 2C. Divide both sides by C and take the logarithm (to any convenient base) of both sides: (t/T)log(3)= log(2), t= T (log(2)/log(3)).

    Here, T= 20 so the time to double is 20 (log(2)/log(3)).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Exponential Growth
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 25th 2009, 09:52 PM
  2. exponential growth
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: November 6th 2008, 07:37 PM
  3. Exponential Growth
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 1st 2008, 08:17 AM
  4. Exponential Growth
    Posted in the Business Math Forum
    Replies: 1
    Last Post: September 10th 2008, 02:54 PM
  5. exponential growth help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 15th 2007, 04:16 PM

Search Tags


/mathhelpforum @mathhelpforum