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The rate of change
(dP)/(dt) of the number of people infected by a disease is modeled by the following differential equation:

(dP)/(dt)=(45)/(125404)P(800-P)
At
t=0, the number of people infected by the disease is 214 and is increasing at a rate of 45 people per hour. What is the limiting value for the total number of people infected by the disease as time increases?
Answer:

The rate of change $\frac{d P}{d t}$ of the number of people infected by a disease is modeled by the following differential equation: $\newline$ $\frac{d P}{d t}=\frac{45}{125404} P(800-P)$ $\newline$ At $t=0$ , the number of people infected by the disease is $214$ and is increasing at a rate of $45$ people per hour. What is the limiting value for the total number of people infected by the disease as time increases? $\newline$ Answer:

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Find instantaneous rates of change

Full solution

Q. The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{45}{125404} P(800-P)

\newline

At

t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

\newline

Answer:

Analyze Differential Equation: To find the limiting value for the total number of people infected by the disease as time increases, we need to analyze the given differential equation: $\newline$ $\frac{dP}{dt} = \frac{45}{125404}P(800-P)$ $\newline$ This equation is a logistic growth model, where the growth rate of the population $P$ is proportional to both the current population $P$ and the difference between the current population and the maximum population ( $800$ in this case). The limiting value, also known as the carrying capacity, is the value of $P$ at which the growth rate $\frac{dP}{dt}$ becomes zero.
Find Growth Rate: Setting the growth rate $(dP)/(dt)$ to zero, we can find the limiting value: $\newline$ $0 = \frac{45}{125404}P(800-P)$ $\newline$ This equation will be zero when either $P = 0$ or when $P = 800$ . Since we are looking for the limiting value as time increases and given that the initial number of infected people is $214$ , which is already greater than zero, we can disregard the solution $P = 0$ .
Calculate Limiting Value: The remaining solution to the equation is $P = 800$ . This is the carrying capacity of the model, which means that as time goes on, the number of infected people will approach this value. This is the limiting value for the total number of people infected by the disease as time increases.

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