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8 October, 18:28

Extreme heat applied to a colony of microorganisms causes the size P of the colony, measured in grams, to decrease according to the exponential decay model dP/dt=-0.4P, where the time t is measured in hours. The size Q of a second colony of microorganisms, also measured in grams, decreases at the constant rate of 1 gram per hour according to the linear model dQ/dt=-1. If at time t=0 the first colony has size P (0) = 2 and the second colony has size Q (0) = 3, at what time will both colonies have the same size?

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  1. 8 October, 18:49
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    Step-by-step explanation:

    dP/dt = - 0.4P

    Separate the variables and integrate.

    dP/P = - 0.4 dt

    ln|P| = - 0.4t + C

    P = Ce^ (-0.4t)

    Use initial condition to find C.

    2 = C^ (0)

    C = 2

    P = 2e^ (-0.4t)

    dQ/dt = - 1

    Integrate.

    Q = - t + C

    Use initial condition to find C.

    3 = 0 + C

    C = 3

    Q = - t + 3

    Set the two equal.

    2e (-0.4t) = - t + 3

    Use a calculator to find t.

    t = 2.156

    The populations have the same size after 2.156 hours.
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