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Homework Help: Calc 2 Exponential Growth(differential equation)

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data

    A tissue culture grows until it has an area of 9 cm^2. Let A(t) be the
    area of the tissue at time t. A model of the growth rate is that:
    A'(t) = k*(sqrt(A(t)) * (9-A(t))

    a. Without solving the equation, show that the maximum rate of growth
    occurs at any time when A(t) = 3 cm^2.

    b. Assume that k = 6. Find the solution corresponding to A(0) = 1 and
    sketch its graph.

    c. Do the same for A(0) = 4.



    2. Relevant equations
    y(t) = b + e^kt ?


    3. The attempt at a solution

    For part A, I took the derivative and set that equal to 0, which gave me back 3.

    For part B, i took it as a separable differential equation and got
    (1/3)(ln((9-A)/(3-sqrt(A))^2)) = kt + C
    but i have no idea how to solve that for A

    Thanks for any help
     
  2. jcsd
  3. Mar 31, 2010 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Well,

    [tex]\ln\left(\frac{9-A}{(3-\sqrt{A})^2}\right)=3kt+3C\implies \frac{9-A}{(3-\sqrt{A})^2}=Be^{3kt}[/tex]

    where [itex]B\equiv e^{3C}[/itex]

    From there, multiply both sides by [itex](3-\sqrt{A})^2[/itex], expand everything out and collect terms in powers of [itex]A[/itex]...you will be left with an equation that is quadratic in terms of [itex]\sqrt{A}[/itex], and I'm sure you know how to solve quadratic equations.
     
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