The rate of change (dP)/(dt) of the number of yeast in a tank is modeled by the following differential equation: (dP)/(dt)=(3768)/(20227)P(1-(P)/( 942)) At t=0, the number of yeast in the tank is 226 and is increasing at a rate of 32 yeast per minute. What is the limiting value for the total number of yeast in the tank as time increases? Answer:

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The rate of change  (dP)/(dt) of the number of yeast in a tank is modeled by the following differential equation:  (dP)/(dt)=(3768)/(20227)P(1-(P)/( 942)) At  t=0, the number of yeast in the tank is 226 and is increasing at a rate of 32 yeast per minute. What is the limiting value for the total number of yeast in the tank as time increases? Answer:

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Logistic Growth Model: The differential equation given is: (dP)/(dt) = (3768)/(20227)P(1 - (P)/(942))  This equation is a logistic growth model, where P is the population of yeast at time t, and (dP)/(dt) is the rate of change of the population with respect to time.  The limiting value, also known as the carrying capacity, is the value of P at which the growth rate (dP)/(dt) becomes zero. This happens when the term (1 - (P)/(942)) equals zero.  Let's find the limiting value by setting the term inside the parentheses to zero and solving for P. 1 - (P)/(942) = 0 P = 942Finding the Limiting Value: The value of P we found, 942, represents the maximum number of yeast the tank can sustain. This is the limiting value or carrying capacity of the yeast population in the tank.  Since the differential equation models logistic growth, as time goes to infinity, the number of yeast will approach this limiting value.  We can confirm this by looking at the structure of the differential equation. As P approaches 942, the term (1 - (P)/(942)) approaches zero, which makes the rate of change (dP)/(dt) approach zero. This indicates that the population growth slows down and stabilizes at the limiting value.

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