Results 1 to 4 of 4

Math Help - problem with ODE

  1. #1
    Member
    Joined
    Nov 2005
    Posts
    111

    problem with ODE

    Hi please can I have some help with this problem:

    An advertising company designs a campaign to introduce a new product to a metropolitan area of population 4 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 10% of the people were aware of the product after 30 days of advertising. What is the number of people who become aware of the product at time t .

    I can not set up the equation...

    My problem I think is thie sentence:
    Suppose that P increases at a rate proportional to the number of people still unaware of the product
    I found something like
    dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

    Thank you

    B
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by braddy
    Hi please can I have some help with this problem:

    An advertising company designs a campaign to introduce a new product to a metropolitan area of population 4 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 10% of the people were aware of the product after 30 days of advertising. What is the number of people who become aware of the product at time t .

    I can not set up the equation...

    My problem I think is thie sentence:
    Suppose that P increases at a rate proportional to the number of people still unaware of the product
    I found something like
    dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

    Thank you

    B

    Looks OK to me, that is exactly what the problem says, and is one of
    the common models for this sort of problem.

    It essentially means some thing like:

    Everyone not aware of the product has an equal chance of becoming aware
    of it in unit time, so the infection rate is proportional to the number not
    infected. Then we treat the population as though its a continuous variable
    rather than discrete, which give us your ODE:

    \frac{dP}{dt}=k(P_0-P),

    with initial condition P(0)=0.

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Quote Originally Posted by braddy
    Hi please can I have some help with this problem:

    An advertising company designs a campaign to introduce a new product to a metropolitan area of population 4 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 10% of the people were aware of the product after 30 days of advertising. What is the number of people who become aware of the product at time t .

    I can not set up the equation...

    My problem I think is thie sentence:
    Suppose that P increases at a rate proportional to the number of people still unaware of the product
    I found something like
    dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

    Thank you

    B
    Here is one way.

    If P(t) = number of people who become aware after time t, then,
    [4 -P(t)] = number of people still unaware.
    So,
    dP(t) /dt = k[4 -P(t)] -----------(i)
    where
    ---P(t) is read "P of t", or "P as a function of t", in Millions of persons.
    ---t is time, in number of days.
    ---k is constant of proportionality.

    For less confusion, let P = P(t), then (i) becomes
    dP/dt = k*(4-P) ---------------------------------------(ia)
    dP = k*(4-P)*dt
    dP/(4-P) = k*dt
    Integrate both sides,
    -ln(4-P) = k*t +C ---------------(ii)

    The problem says:
    ---When t=0, P(0) = none = 0 also. ----------------------(1)
    ---when t=30days, P(30) = 10% of 4Million = 0.4M --------(2)

    Apply condition (1) into (ii),
    -ln(4-0) = k*0 +C
    So, C = -ln(4) -----------------***

    Apply that, and condition (2), into (ii),
    -ln(4 -0.4) = k*30 -ln(4)
    Simplifying,
    ln(4) -ln(3.6) = 30k
    ln[4/3.6] = 30k
    k = (1/30)ln(1/0.9) = (1/30)[ln(1) -ln(0.9)]
    k = -(1/30)ln(0.9)-----------------------------------***

    To find P in general, going back to (ii),
    -ln(4-P) = kt -ln(4)
    ln(4) -ln(4-P) = kt
    ln[4/(4-P)] = kt
    4/(4-P) = e^(kt)
    The reciprocal of the LHS = the reciprocal of the RHS,
    (4-P)/4 = e^(-kt)
    4-P = 4e^(-kt)
    P = 4 -4e^(-kt)
    P = 4[1 -e^(-kt)] ----------------------answer.

    where k = -(1/30)ln(0.9), so,
    P = 4[1 -e^(ln(0.9) *t/30)] --------------answer.

    If we go back to P as function of t, or P(t), and let Po = total population, then,
    P(t) = (Po)[1 -e^(-kt)] ------------answer.
    Etc.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2005
    Posts
    111
    Quote Originally Posted by ticbol
    Here is one way.

    If P(t) = number of people who become aware after time t, then,
    [4 -P(t)] = number of people still unaware.
    So,
    dP(t) /dt = k[4 -P(t)] -----------(i)
    where
    ---P(t) is read "P of t", or "P as a function of t", in Millions of persons.
    ---t is time, in number of days.
    ---k is constant of proportionality.

    For less confusion, let P = P(t), then (i) becomes
    dP/dt = k*(4-P) ---------------------------------------(ia)
    dP = k*(4-P)*dt
    dP/(4-P) = k*dt
    Integrate both sides,
    -ln(4-P) = k*t +C ---------------(ii)

    The problem says:
    ---When t=0, P(0) = none = 0 also. ----------------------(1)
    ---when t=30days, P(30) = 10% of 4Million = 0.4M --------(2)

    Apply condition (1) into (ii),
    -ln(4-0) = k*0 +C
    So, C = -ln(4) -----------------***

    Apply that, and condition (2), into (ii),
    -ln(4 -0.4) = k*30 -ln(4)
    Simplifying,
    ln(4) -ln(3.6) = 30k
    ln[4/3.6] = 30k
    k = (1/30)ln(1/0.9) = (1/30)[ln(1) -ln(0.9)]
    k = -(1/30)ln(0.9)-----------------------------------***

    To find P in general, going back to (ii),
    -ln(4-P) = kt -ln(4)
    ln(4) -ln(4-P) = kt
    ln[4/(4-P)] = kt
    4/(4-P) = e^(kt)
    The reciprocal of the LHS = the reciprocal of the RHS,
    (4-P)/4 = e^(-kt)
    4-P = 4e^(-kt)
    P = 4 -4e^(-kt)
    P = 4[1 -e^(-kt)] ----------------------answer.

    where k = -(1/30)ln(0.9), so,
    P = 4[1 -e^(ln(0.9) *t/30)] --------------answer.

    If we go back to P as function of t, or P(t), and let Po = total population, then,
    P(t) = (Po)[1 -e^(-kt)] ------------answer.
    Etc.

    Thank you very much!
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum