The rate of change (dP)/(dt) 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. (dP)/(dt)=◻

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The rate of change  (dP)/(dt) 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.  (dP)/(dt)=◻

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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: $28 = r \cdot 184 \left(1 - \frac{184}{896}\right)$Substitute Value for r: First, calculate the fraction of the carrying capacity that the current population represents: $\frac{184}{896} = \frac{23}{112}$Simplify Expression: Now, substitute this value back into the equation to solve for $r$: $28 = r \cdot 184 \left(1 - \frac{23}{112}\right)$Divide Both Sides: Simplify the expression inside the parentheses: $1 - \frac{23}{112} = \frac{112}{112} - \frac{23}{112} = \frac{89}{112}$Perform Division: Now, the equation becomes: $28 = r \cdot 184 \cdot \frac{89}{112}$Calculate r Value: To solve for $r$, divide both sides of the equation by $184 \cdot \frac{89}{112}$: $r = \frac{28}{184 \cdot \frac{89}{112}}$Simplify Fraction: Perform the division to find $r$: $r = \frac{28 \cdot 112}{184 \cdot 89}$Find r Value: Calculate the value of $r$: $r = \frac{3136}{16376}$Approximate r Value: Simplify the fraction to find $r$: $r = \frac{1}{\frac{16376}{3136}}$Write Differential Equation: Calculate the denominator of the fraction: $\frac{16376}{3136} = 5.22$ (approximately)Now, find the value of $r$: $r \approx \frac{1}{5.22}$The approximate value of $r$ is: $r \approx 0.1916$ (rounded to four decimal places)Now that we have the value of $r$, we can write the logistic differential equation for the fox population: $\frac{dP}{dt} = 0.1916P\left(1 - \frac{P}{896}\right)$

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