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 720 students. At 3PM, the number of students who heard the rumor is 115 and is increasing at a rate of 47 students per hour. Write a differential equation to describe the situation. (dP)/(dt)=◻" Submit Answer "

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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 720 students. At  3PM, the number of students who heard the rumor is 115 and is increasing at a rate of 47 students per hour. Write a differential equation to describe the situation.  (dP)/(dt)=◻" Submit Answer "

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Logistic Differential Equation: The logistic differential equation is commonly used to model population growth that is restricted by limited resources, or in this case, the maximum capacity of the school. The general form of the logistic equation is:  $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$  where: - $ \frac{dP}{dt} $ is the rate of change of the population (or the number of students who heard the rumor in this context), - $ r $ is the intrinsic growth rate, - $ P $ is the current population (or the 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 720 students, and at 3 PM, $ P = 115 $ students have heard the rumor. The rate of change $ \frac{dP}{dt} $ at that time is 47 students per hour. We need to find the value of $ r $ that makes the equation true for these values.Find Intrinsic Growth Rate: To find the intrinsic growth rate $ r $, we can use the given rate of change when $ P = 115 $:  $$47 = r \cdot 115 \left(1 - \frac{115}{720}\right)$$  Now we solve for $ r $:  $$r = \frac{47}{115 \left(1 - \frac{115}{720}\right)}$$Calculate Denominator: We calculate the denominator of the fraction:  $$115 \left(1 - \frac{115}{720}\right) = 115 \left(\frac{720 - 115}{720}\right) = 115 \left(\frac{605}{720}\right)$$  Now we calculate the value of $ r $:  $$r = \frac{47}{115 \cdot \frac{605}{720}}$$Simplify Fraction: We simplify the fraction:  $$r = \frac{47 \cdot 720}{115 \cdot 605}$$  Now we perform the multiplication and division to find $ r $:  $$r = \frac{33840}{69475}$$  $$r \approx 0.487$$Write Differential Equation: Now that we have the value of $ r $, we can write the logistic differential equation for this situation:  $$\frac{dP}{dt} = 0.487P\left(1 - \frac{P}{720}\right)$$  This is the differential equation that describes the rate of change of the number of students who heard the rumor at the school.

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