Math Assignment HELP!!?

Question 3. Let V (t) be the volume of a benign tumour in cm3 after t years. For t greater than equal to 0, suppose that V (t) satisfies the following differential equation

dV/dt =(1+t)e−t

a) If initially V (0) = 1, find V (t).

b) Compute lim V (t) and interpret. t→∞

c) Use Newton’s method to find when the volume of the tumour will be 2 cm^3. Use 5 decimal places in your computations and find the answer with 3 decimal places of precision.

1 Answer

  • hfshaw
    Lv 7
    7 years ago
    Favorite Answer

    You have (I think):

    dV/dt = (1 + t)*exp(-t)

    This is a simple, separable equation:

    dV = (exp(-t) + t*exp(-t)) dt

    Integrate both sides:

    V(t) = -exp(-t) - (t+1)*exp(-t) + c

    where c is the constant of integration.

    V(t) = c - (t + 2)*exp(-t)

    Now use the initial condition to solve for the constant:

    V(0) = 1 = c - 2

    c = 3

    So the solution to this initial value problem is:

    V(t) = 3 - (t + 2)*exp(-t)

    As t→∞, the exponential term goes to zero, so the long-term (steady state) volume of the tumor is 3

    For the final part of the question, you are trying to find the value of t such that:

    2 = 3 - (t + 2)*exp(-t)

    1 = (t + 2)*exp(-t)

    exp(t) = t + 2

    This is a transcendental equation that cannot be solved algebraically. You can do the numerical approximations needed to find when V(t) = 2; they are not amenable to writing out in this medium. Personally, I would simply do it by trial and error using a spreadsheet or a calculator. You should find that t = ~1.146 yrs

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