The rate of change (dP)/(dt) of the number of people infected by a disease is modeled by a logistic differential equation. The maximum capacity of the village is 943 people. At 8 PM, the number of people infected by the disease is 106 and is increasing at a rate of 30 people per hour. Write a differential equation to describe the situation. (dP)/(dt)=◻

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The rate of change  (dP)/(dt) of the number of people infected by a disease is modeled by a logistic differential equation. The maximum capacity of the village is 943 people. At 8 PM, the number of people infected by the disease is 106 and is increasing at a rate of 30 people per hour. Write a differential equation to describe the situation.  (dP)/(dt)=◻

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Logistic Differential Equation: The logistic differential equation is generally given by the formula: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$ where: - $P$ is the current population (number of people infected), - $r$ is the growth rate (rate of increase of the infection), - $K$ is the carrying capacity (maximum capacity of the village), - $\frac{dP}{dt}$ is the rate of change of the population (number of people infected) over time.Given Carrying Capacity: We are given the carrying capacity $K = 943$ people.Given Population and Rate of Change: We are also given that at a certain time (8 PM), the number of people infected $P = 106$ and the rate of change of the number of people infected $\frac{dP}{dt} = 30$ people per hour.Finding Growth Rate: To find the growth rate $r$, we can use the given rate of change when $P = 106$: $$30 = r \cdot 106 \left(1 - \frac{106}{943}\right)$$ Now we solve for $r$: $$r = \frac{30}{106 \left(1 - \frac{106}{943}\right)}$$Calculating Growth Rate: Calculate the value of $r$: $$r = \frac{30}{106 \left(1 - \frac{106}{943}\right)}$$ $$r = \frac{30}{106 \left(\frac{943 - 106}{943}\right)}$$ $$r = \frac{30}{106 \left(\frac{837}{943}\right)}$$ $$r = \frac{30 \cdot 943}{106 \cdot 837}$$ $$r \approx \frac{28329}{88962}$$ $$r \approx 0.3185$$ (rounded to four decimal places)Writing Logistic Differential Equation: Now we can write the logistic differential equation using the value of $r$ and the carrying capacity $K$: $$\frac{dP}{dt} = 0.3185P\left(1 - \frac{P}{943}\right)$$

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