The rate of change (dP)/(dt) of the number of students who heard a rumor is modeled by a logistic differential equation. The maximum capacity of the school is 861 students. At 2AM, the number of students who heard the rumor is 213 and is increasing at a rate of 34 students per hour. Write a differential equation to describe the situation. (dP)/(dt)=◻

Question

The rate of change  (dP)/(dt) of the number of students who heard a rumor is modeled by a logistic differential equation. The maximum capacity of the school is 861 students. At  2AM, the number of students who heard the rumor is 213 and is increasing at a rate of 34 students 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: - $ \frac{dP}{dt} $ is the rate of change of the population (or in this case, the number of students who heard the rumor), - $ r $ is the intrinsic growth rate, - $ P $ is the current population (or the current number of students who have heard the rumor), - $ K $ is the carrying capacity (or the maximum capacity of the school).  We are given that the maximum capacity $ K $ is 861 students.Find Intrinsic Growth Rate: We need to find the intrinsic growth rate $ r $. We know that at 2AM, the number of students who heard the rumor is 213 and is increasing at a rate of 34 students per hour. This rate of increase is actually $ \frac{dP}{dt} $ when $ P = 213 $.  So, we can plug these values into the logistic differential equation to solve for $ r $: $$ 34 = r \cdot 213 \left(1 - \frac{213}{861}\right) $$Calculate Fraction: First, calculate the fraction of the carrying capacity that has been reached when $ P = 213 $: $$ \frac{213}{861} $$ $$ = \frac{1}{4} $$Solve for r: Now, substitute $ \frac{1}{4} $ back into the equation and solve for $ r $: $$ 34 = r \cdot 213 \left(1 - \frac{1}{4}\right) $$ $$ 34 = r \cdot 213 \cdot \frac{3}{4} $$ $$ 34 = r \cdot 159.75 $$ $$ r = \frac{34}{159.75} $$ $$ r \approx 0.213 $$Write Logistic Differential Equation: Now that we have the value of $ r $, we can write the logistic differential equation for this situation: $$ \frac{dP}{dt} = 0.213P\left(1 - \frac{P}{861}\right) $$

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