The rate of change (dP)/(dt) of the number of people on an island is modeled by a logistic differential equation. The maximum capacity of the island is 745 people. At 11 AM, the number of people on the island is 186 and is increasing at a rate of 24 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 on an island is modeled by a logistic differential equation. The maximum capacity of the island is 745 people. At 11 AM, the number of people on the island is 186 and is increasing at a rate of 24 people per hour. Write a differential equation to describe the situation.  (dP)/(dt)=◻

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Logistic Growth Model: The logistic growth model is given by the differential equation $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$, where $P$ is the population at time $t$, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity of the environment.Given Carrying Capacity: We are given the carrying capacity $K = 745$ people. This will be used in our differential equation.Find Intrinsic Growth Rate: We need to find the intrinsic growth rate $r$. We know that at 11 AM, the population $P = 186$ and the rate of change $\frac{dP}{dt} = 24$ people per hour. We can use the logistic growth formula to solve for $r$.Substitute Values: Substitute $P = 186$, $\frac{dP}{dt} = 24$, and $K = 745$ into the logistic growth formula to find $r$: $24 = r \cdot 186 \left(1 - \frac{186}{745}\right)$.Calculate r: Calculate the value of $r$: $24 = r \cdot 186 \left(\frac{745 - 186}{745}\right)$, $24 = r \cdot 186 \cdot \frac{559}{745}$, $24 = r \cdot \frac{186 \cdot 559}{745}$, $r = \frac{24 \cdot 745}{186 \cdot 559}$.Final Logistic Differential Equation: Perform the calculation to find $r$: $r = \frac{24 \cdot 745}{186 \cdot 559}$, $r \approx \frac{17880}{103914}$, $r \approx 0.172$.Now that we have the value of $r$, we can write the logistic differential equation: $\frac{dP}{dt} = 0.172P\left(1 - \frac{P}{745}\right)$.

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