The rate of change (dP)/(dt) of the number of people at a carnival is modeled by the following differential equation: (dP)/(dt)=(2881)/(7178)P(1-(P)/( 536)) At t=0, the number of people at the carnival is 148 and is increasing at a rate of 43 people per hour. At what value of P does the graph of P(t) have an inflection point? Answer:

Question

The rate of change  (dP)/(dt) of the number of people at a carnival is modeled by the following differential equation:  (dP)/(dt)=(2881)/(7178)P(1-(P)/( 536)) At  t=0, the number of people at the carnival is 148 and is increasing at a rate of 43 people per hour. At what value of  P does the graph of  P(t) have an inflection point? Answer:

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Find Second Derivative: To find the inflection point, we need to find the second derivative of P with respect to t and then determine when it changes sign. The given differential equation is:  (dP)/(dt) = (2881)/(7178)P(1 - (P)/(536))  First, we need to find the first derivative of (dP)/(dt) with respect to P, which will give us (d^2P)/(dt^2).Calculate First Derivative: The first derivative of (dP)/(dt) with respect to P is given by the derivative of the right-hand side of the equation with respect to P. We apply the product rule and the chain rule to find this derivative:  (d^2P)/(dt^2) = (d)/(dP) [(2881)/(7178)P(1 - (P)/(536))]  Let's denote (2881)/(7178) as a constant k for simplicity:  k = (2881)/(7178)  Now, we differentiate:  (d^2P)/(dt^2) = k * (d)/(dP) [P - (P^2)/(536)]  (d^2P)/(dt^2) = k * (1 - (2P)/(536))Set Second Derivative to Zero: To find the inflection point, we set the second derivative equal to zero and solve for P:  0 = k * (1 - (2P)/(536))  Since k is a positive constant, we can divide both sides by k without changing the inequality:  0 = 1 - (2P)/(536)  Now, we solve for P:  (2P)/(536) = 1  2P = 536  P = 536 / 2  P = 268Solve for Inflection Point: We have found that the second derivative changes sign at P = 268. This is the value of P at which the graph of P(t) has an inflection point.

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