Answer to Question #280559 in Differential Equations for lev

Question #280559

a bacteria culture is known to grow at a rate proportional to the amount present. after one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 30000 strands. Find an expression fo the approximate number of strands of the bacteria present in the culture at any time t.


1
Expert's answer
2021-12-17T13:29:55-0500

Let P(t)P(t) be the size of the population of the bacteria culture at any time t.t.

Then

dPdt=kP\dfrac{dP}{dt}=kP

dPP=kdt\dfrac{dP}{P}=kdt

P(t)=P(0)ektP(t)=P(0)e^{kt}

Given P(1)=1000,P(4)=30000P(1)=1000, P(4)=30000


P(4)P(1)=P(0)e4kP(0)ek=300001000\dfrac{P(4)}{P(1)}=\dfrac{P(0)e^{4k}}{P(0)e^{k}}=\dfrac{30000}{1000}

e3k=30e^{3k}=30

k=ln303k=\dfrac{\ln 30}{3}

P(t)=P(0)e(ln30/3)tP(t)=P(0)e^{(\ln 30/3)t}

P(t)=P(0)(30)t/3P(t)=P(0)(30)^{t/3}

P(1)=P(0)(30)1/3=1000P(1)=P(0)(30)^{1/3}=1000

P(0)=1000303P(0)=\dfrac{1000}{\sqrt[3]{30}}

P(t)=1000303(30)t/3P(t)=\dfrac{1000}{\sqrt[3]{30}}(30)^{t/3}

P(t)322(30)t/3,t0P(t)\approx322(30)^{t/3}, t\geq 0


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