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Math Help - optimization problems involving exponential functions

  1. #1
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    optimization problems involving exponential functions

    Please help, thank you!

    1. a colony of bacteria in a culture grows at a rate given by N(t)=2^(t/5) where N is the number of bacteria t minutes from the beginning. The colony is allowed to grow for 60 mins at which time a drug is introduced to kill the bacteria. The number of bacteria killed is given by K(t)=e^(t/3) where K bacteria are killed at time t minutes.
    a) determine the max number of bacteria present and the time at which this occurs.
    b) determine the time at which the bacteria colony is obliterated.


    2. Loraine is studying for 2 different exams. Because of the nature of the courses, the measure of study effectiveness on a scale from 0 to 10 for the first course is E1 = 0.6(9 + te^-t/20), while the measure for the 2nd course is E2=0.5(10 + te^-t/10). Loraine is prepared to spend up to 30 hours in total studying. THe total effectiveness is given by f(t)=E1+E2. How should this time be allotted to maximize total effectiveness?
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  2. #2
    Senior Member apcalculus's Avatar
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    re:

    Quote Originally Posted by checkmarks View Post
    Please help, thank you!

    1. a colony of bacteria in a culture grows at a rate given by N(t)=2^(t/5) where N is the number of bacteria t minutes from the beginning. The colony is allowed to grow for 60 mins at which time a drug is introduced to kill the bacteria. The number of bacteria killed is given by K(t)=e^(t/3) where K bacteria are killed at time t minutes.
    a) determine the max number of bacteria present and the time at which this occurs.
    b) determine the time at which the bacteria colony is obliterated.


    2. Loraine is studying for 2 different exams. Because of the nature of the courses, the measure of study effectiveness on a scale from 0 to 10 for the first course is E1 = 0.6(9 + te^-t/20), while the measure for the 2nd course is E2=0.5(10 + te^-t/10). Loraine is prepared to spend up to 30 hours in total studying. THe total effectiveness is given by f(t)=E1+E2. How should this time be allotted to maximize total effectiveness?
    For problem 1), the net change in the number of bacteria between t=0 and t=x is given by:

    T(x) - T(0) = INTEGRAL(RATE of GROWTH - RATE of REMOVAL) from 0 to x

    so T(x) = T(0) + INTEGRAL(RATE of GROWTH - RATE of REMOVAL) from 0 to x

    To optimize the total number of bacteria, we differentiate (using FTC here) and get

    T'(x) = RateOfGrowth(x) - RateOfRemoval(x)
    Find the critical number and justify that it's an optimum of the type you want (i.e. max and not min)

    for part b)
    Set up the integral as above with T(x) = 0, then solve for x. So
    - T(0) = INTEGRAL(RATE of GROWTH - RATE of REMOVAL) from 0 to x

    Good luck!!
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