The rate of change $\frac{d P}{d t}$ of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is $896$ fox. At $8$ PM, the number of fox at the national park is $184$ and is increasing at a rate of $28$ fox per day. Write a differential equation to describe the situation. $\newline$ $\frac{d P}{d t}=\square$

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Q. The rate of change

\frac{d P}{d t}

of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

896

fox. At

8

PM, the number of fox at the national park is

184

and is increasing at a rate of

28

fox per day. Write a differential equation to describe the situation.

\newline

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

Logistic Growth Model: The logistic growth model can be represented 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 $K = 896$ foxes. This is the maximum number of foxes that the park can support.
Initial Population Data: We are also given that at $8$ PM, the number of foxes $P = 184$ and the rate of change of the population $\frac{dP}{dt} = 28$ foxes per day.
Calculate Fraction: To find the intrinsic growth rate $r$ , we can use the given rate of change and the current population to solve for $r$ in the logistic growth equation. Plugging in the values we have: $\newline$ $28 = r \cdot 184 \left(1 - \frac{184}{896}\right)$
Substitute Value for r: First, calculate the fraction of the carrying capacity that the current population represents: $\newline$ $\frac{184}{896} = \frac{23}{112}$
Simplify Expression: Now, substitute this value back into the equation to solve for $r$ : $\newline$ $28 = r \cdot 184 \left(1 - \frac{23}{112}\right)$
Divide Both Sides: Simplify the expression inside the parentheses: $\newline$ $1 - \frac{23}{112} = \frac{112}{112} - \frac{23}{112} = \frac{89}{112}$
Perform Division: Now, the equation becomes: $\newline$ $28 = r \cdot 184 \cdot \frac{89}{112}$
Calculate r Value: To solve for $r$ , divide both sides of the equation by $184 \cdot \frac{89}{112}$ : $\newline$ $r = \frac{28}{184 \cdot \frac{89}{112}}$
Simplify Fraction: Perform the division to find $r$ : $\newline$ $r = \frac{28 \cdot 112}{184 \cdot 89}$
Find r Value: Calculate the value of $r$ : $\newline$ $r = \frac{3136}{16376}$
Approximate r Value: Simplify the fraction to find $r$ : $\newline$ $r = \frac{1}{\frac{16376}{3136}}$
Write Differential Equation: Calculate the denominator of the fraction: $\newline$ $\frac{16376}{3136} = 5.22$ (approximately)
Write Differential Equation: Calculate the denominator of the fraction: $\newline$ $\frac{16376}{3136} = 5.22$ (approximately)Now, find the value of $r$ : $\newline$ $r \approx \frac{1}{5.22}$
Write Differential Equation: Calculate the denominator of the fraction: $\newline$ $\frac{16376}{3136} = 5.22$ (approximately)Now, find the value of $r$ : $\newline$ $r \approx \frac{1}{5.22}$ The approximate value of $r$ is: $\newline$ $r \approx 0.1916$ (rounded to four decimal places)
Write Differential Equation: Calculate the denominator of the fraction: $\newline$ $\frac{16376}{3136} = 5.22$ (approximately)Now, find the value of $r$ : $\newline$ $r \approx \frac{1}{5.22}$ The approximate value of $r$ is: $\newline$ $r \approx 0.1916$ (rounded to four decimal places)Now that we have the value of $r$ , we can write the logistic differential equation for the fox population: $\newline$ $\frac{dP}{dt} = 0.1916P\left(1 - \frac{P}{896}\right)$