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22 June, 02:43

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. after 3 hours it is observed that 400 bacteria are present. after 10 hours 2000 bacteria are present. what was the initial number of bacteria? (round your answer to the nearest integer.)

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  1. 22 June, 02:45
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    The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. = >

    dx/dt=kt

    Integrate, x (t) = kt^2/2+C ... (0)

    Substitute x (3) = 400 = > k (3^2) / 2+C=400 = > 4.5k+C=400 ... (1)

    Substitute x (10) = 2000 = > k (10^2)) / 2+C=2000 = > 50k+C=2000 ... (2)

    (2) - (1) = > 45.5k=1600 = > k=35.165 ... (3)

    substitute (3) in (1)

    4.5 (35.165) + C=400 = > C=241.76 ... (4)

    Substitute (3) and (4) in (0) to get

    x (t) = kt^2/2+C=17.582t^2+241.76 ... (5)

    So at t=0, x (0) = 241.7 or 242 (to the nearest bacterium)
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