Results 1 to 2 of 2

Thread: growth and decay

  1. #1
    Newbie
    Joined
    Oct 2007
    Posts
    14

    growth and decay

    I need help finding the formula for P(t)!!


    A population P=P(t) grows exponentially with P(3)=60 and P(6)=90. Find the formula for P(t), and find the population when t=10.

    I know how to do this but only when P(0) is given, which it is not in this case. Please give some hints = )
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, redpanda11!

    A population $\displaystyle P(t)$ grows exponentially with: $\displaystyle P(3)=60$ and $\displaystyle P(6)=90$
    Find the formula for $\displaystyle P(t)$
    The function has the form: .$\displaystyle P(t) \;=\;ae^{bt}$

    From $\displaystyle P(3) = 60$, we have: .$\displaystyle ae^{3b} \:=\:60$ . [1]

    From $\displaystyle P(6)=90$, we have: .$\displaystyle ae^{6b} \:=\:90$ . [2]

    Divide [2] by [1]: .$\displaystyle \frac{ae^{6b}}{ae^{3b}} \:=\:\frac{90}{60}\quad\Rightarrow\quad e^{3b} \:=\:1.5\quad\Rightarrow\quad 3b \:=\:\ln(1.5)$

    .Hence: .$\displaystyle \boxed{b \:=\:\frac{1}{3}\ln(1.5)}$


    The function (so far) is: .$\displaystyle P(t) \;=\;ae^{\left(\frac{1}{3}\ln15\right)t} \;=\;a\left(e^{\ln15}\right)^{\frac{1}{3}t}\quad\R ightarrow\quad P(t)\:=\:a(1.5)^{\frac{1}{3}t} $

    From $\displaystyle P(3) = 60$, we have: .$\displaystyle a(1.5)^1 \:=\:60\quad\Rightarrow\quad 1.5a \:=\:60\quad\Rightarrow\quad\boxed{a \:=\:40}$


    Therefore: .$\displaystyle {\color{blue}\boxed{P(t) \;=\;40(1.5)^{\frac{1}{3}t}}} $

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Growth & Decay
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 18th 2010, 07:40 AM
  2. Decay and Growth
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Nov 4th 2009, 03:46 PM
  3. Exponents Growth and Decay
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Oct 21st 2009, 06:59 PM
  4. Growth and Decay
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 8th 2009, 07:18 PM
  5. Exponential Growth & Decay
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 11th 2007, 08:33 PM

Search Tags


/mathhelpforum @mathhelpforum