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17 December, 19:56

A bacteria culture starts with 880 bacteria and grows at a rate proportional to its size. after 5 hours there will be 4400 bacteria. (a) express the population after t hours as a function of t. population: (function of t) (b) what will be the population after 2 hours? (c) how long will it take for the population to reach 2550?

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  1. 17 December, 20:03
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    Don't know whether or not you've encountered differential equations yet, but will try that approach here.

    The growth rate is dy/dt = ky (which states that the rate is proportional to the size of the population, y, and k is a constant.

    Grouping like terms,

    dy

    - - - = kt, so y = Ne^kt

    y

    We are told that at t=0, there are 880 bacteria. Thus, 880=N. Therefore,

    y = 880e^ (kt). After 5 hours the pop will be 4400; using this info, find k:

    4400=880e^ (5k), or 5 = e^ (5k). So, our y = 880e^ (kt) becomes

    y = 880e^ (5t).

    What will be the pop after 2 hours? y (2) = 880e^ (10) = 880 (22026) =

    approx. 19,383,290 bacteria

    Time to reach a pop of 2550? 2550 = 880e^ (5t). Find t.

    ln 2550 = ln 880 + 5t, so ln 2550 - ln 880 = 5t. Divide both sides by 5 to obtain this time, t.
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