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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)=◻

The rate of change $\frac{d P}{d t}$ 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. $\newline$ $\frac{d P}{d t}=\square$

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Write exponential functions: word problems

Full solution

Q. The rate of change

\frac{d P}{d t}

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.

\newline

\frac{d P}{d t}=\square

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 $r$ : $\newline$ $24 = r \cdot 186 \left(1 - \frac{186}{745}\right)$ .
Calculate r: Calculate the value of $r$ : $\newline$ $24 = r \cdot 186 \left(\frac{745 - 186}{745}\right)$ , $\newline$ $24 = r \cdot 186 \cdot \frac{559}{745}$ , $\newline$ $24 = r \cdot \frac{186 \cdot 559}{745}$ , $\newline$ $r = \frac{24 \cdot 745}{186 \cdot 559}$ .
Final Logistic Differential Equation: Perform the calculation to find $r$ : $\newline$ $r = \frac{24 \cdot 745}{186 \cdot 559}$ , $\newline$ $r \approx \frac{17880}{103914}$ , $\newline$ $r \approx 0.172$ .
Final Logistic Differential Equation: Perform the calculation to find $r$ : $\newline$ $r = \frac{24 \cdot 745}{186 \cdot 559}$ , $\newline$ $r \approx \frac{17880}{103914}$ , $\newline$ $r \approx 0.172$ .Now that we have the value of $r$ , we can write the logistic differential equation: $\newline$ $\frac{dP}{dt} = 0.172P\left(1 - \frac{P}{745}\right)$ .

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