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Math Help - logistic growth problem

  1. #1
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    logistic growth problem

    The population of fish in a certain lake at time t months is given by the function:
    P(t) = 20,000/(1+24e^(-t/4)) , where T is greater than or equal to 0. There is an upper limit on the fish population due to the oxygen supply, available food, etc.

    A. What is the initial population of fish?
    B. When will there be 15,000 fish?
    C. What is the maximum number of fish possible in the lake?

    A and B i know how to do, but C is a little confusing. To find the maximum, i took the derivative of P(t), which after simplifying, is 120,000 e^(-t/4) / (1+24e^(-t/4))^2. i tried to find critical numbers by setting the numerator and denominator equal to 0, but on the top i end up with e^(-t/4) = 0 and exponential functions never equal 0 so that's no solution. and on the bottom i get e^(-t/4) = -1/24 and since exponential functions are never negative, that one has no solution either. i saw the answer that said take the limit as t approaches infinite and you get 20,000 as the maximum. but how come i couldn't use the derivative to find the maximum in this problem?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Consider the graph -- it is an increasing function with an inflection pt at

    10,000.

    Your derivative shows this as there are no critical pts and the derivative is positive

    So the max ocurs at infinity
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    Quote Originally Posted by Calculus26 View Post
    Consider the graph -- it is an increasing function with an inflection pt at

    10,000.

    Your derivative shows this as there are no critical pts and the derivative is positive

    So the max ocurs at infinity
    since this is a logistic growth, the bottom of the graph is flat, the graph increases, and then the top of the graph is flat. how come the first derivative test didn't pick up a max or min at P(t) = 0 or P(t) = 20,000? since the graph is flat in those 2 areas, shouldn't the derivative be 0 and thus register as max and mins?
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  4. #4
    MHF Contributor Calculus26's Avatar
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    Don't confuse "looks flat" with horizontal. If the rate of change is small but not 0 the graph appears flat relatively speaking but not horizontal

    The graph is not flat in the sense the derivative is never 0.

    Granted initially the rate of increase is small but not 0 otherwise the population would never increase.

    Similarly as t - > infinity the graph flattens out but again as the derivative is never 0 the population approaches 20,000 asymptotically but again

    as the derivative is not 0 the population is increasing at a very small rate
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  5. #5
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    Quote Originally Posted by oblixps View Post
    The population of fish in a certain lake at time t months is given by the function:
    P(t) = 20,000/(1+24e^(-t/4)) , where T is greater than or equal to 0. There is an upper limit on the fish population due to the oxygen supply, available food, etc.

    A. What is the initial population of fish?
    B. When will there be 15,000 fish?
    C. What is the maximum number of fish possible in the lake?
    \lim_{t \to \infty} P(t) = 20000
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