Answer to Question #204126 in Differential Equations for Lou Andrei Palilio

Question

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

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?

Answer 1

A.

Exponential growth

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

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

$$25=10e^{k(2)}$$

$$2k=\ln(2.5)$$

$$k=\dfrac{\ln(2.5)}{2}$$

$$P(t)=10e^{{\ln(2.5) \over 2}t}$$

Or

$$P(t)=10(2.5)^{t/2}, P(t)\leq100$$

B.

$$P(5)=10(2.5)^{5/2}$$

$$P(5)\approx98.821\ g$$

C.

$$P(t)=10(2.5)^{t/2}=75$$

$$\ln(2.5)=\ln(7.5)$$

$$t=2\cdot\dfrac{\ln(7.5)}{\ln(2.5)}$$

$$t\approx4.398\ h$$