Question #338366

Activity 2






Directions :Solve the logistics problem. show your solution.




The population of a certain bacteria follows the logistic growth pattern, initially, there are 10 g of bacteria present in the culture. Two hours later, the culture weighs 25 g. The maximum weight of the culture is 100g.




a.Write the corresponding logistic model for the bacterial growth?




b. What is the weight of the culture after 5 hours?




c. When will the culture's weight be 75 g?

1
Expert's answer
2022-05-10T08:16:59-0400

A.

Exponential growth



P(t)=P(0)ekt,k>0P(t)=P(0)e^{kt}, k>0

Given P(0)=10 g,P(2)=25 g,P100 gP(0)=10\ g, P(2)=25\ g, P\leq 100\ g

P(t)=10ektP(t)=10e^{kt}25=10ek(2)25=10e^{k(2)}2k=ln(2.5)2k=\ln(2.5)k=ln(2.5)2k=\dfrac{\ln(2.5)}{2}

The corresponding logistic model for the bacterial growth

P(t)=10(2.5)t/2,P(t)100P(t)=10(2.5)^{t/2}, P(t)\le100


B.


P(5)=10(2.5)5/2P(5)=10(2.5)^{5/2}P(5)98.821 gP(5)\approx98.821\ g

C.


P(t)=10(2.5)t/2P(t)=10(2.5)^{t/2}

P(t)=75P(t)=75


75=10(2.5)t/275=10(2.5)^{t/2}ln(2.5)(t2)=ln(7.5)\ln(2.5)\cdot(\dfrac{t}{2})=\ln(7.5)t=2ln(7.5)ln(2.5)t=2\cdot\dfrac{\ln(7.5)}{\ln(2.5)}t4.398 ht\approx4.398\ h




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