The rate of change (dP)/(dt) of the number of students who heard a rumor is modeled by the following differential equation: (dP)/(dt)=(6500)/(19153)P(1-(P)/( 500)) At t=0, the number of students who heard the rumor is 179 and is increasing at a rate of 39 students per hour. Find lim_(t rarr oo)P(t). Answer:

Question

The rate of change  (dP)/(dt) of the number of students who heard a rumor is modeled by the following differential equation:  (dP)/(dt)=(6500)/(19153)P(1-(P)/( 500)) At  t=0, the number of students who heard the rumor is 179 and is increasing at a rate of 39 students per hour. Find  lim_(t rarr oo)P(t). Answer:

Bytelearn · Accepted Answer

Given Differential Equation: The given differential equation is a logistic growth model, which is commonly used to describe population growth that is limited by resources or other factors. The equation is:  (dP)/(dt)=(6500)/(19153)P(1-(P)/(500))  This equation suggests that the growth rate of the rumor (dP/dt) depends on the current number of students who have heard the rumor (P), with a carrying capacity of 500 students. The carrying capacity is the maximum number of students that can hear the rumor given the constraints of the environment, in this case, the school.  To find the limit of P(t) as t approaches infinity, we need to understand the behavior of the logistic growth model. As t increases, P(t) will grow until it approaches the carrying capacity, because the term (1 - (P/500)) will approach zero as P approaches 500, causing the growth rate (dP/dt) to approach zero.  Therefore, the limit of P(t) as t approaches infinity is the carrying capacity of the model, which is 500 students.

Full solution

More problems from Interpreting Linear Expressions

Related topics