The rate of change (dP)/(dt) of the number of students who heard a rumor is modeled by the following differential equation: (dP)/(dt)=(379)/(4760)P(1-(P)/( 758)) At t=0, the number of students who heard the rumor is 238 and is increasing at a rate of 13 students per hour. At what value of P is P(t) growing the fastest? 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)=(379)/(4760)P(1-(P)/( 758)) At  t=0, the number of students who heard the rumor is 238 and is increasing at a rate of 13 students per hour. At what value of  P is  P(t) growing the fastest? Answer:

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Given Differential Equation: The given differential equation is: (dP)/(dt) = (379)/(4760)P(1 - (P)/(758)) To find the value of P where P(t) is growing the fastest, we need to find the maximum point of the rate of change function (dP)/(dt). This occurs when the derivative of (dP)/(dt) with respect to P is equal to zero.Derivative Setup: Let's set up the derivative of (dP)/(dt) with respect to P: d/dP [(dP)/(dt)] = d/dP [(379)/(4760)P(1 - (P)/(758))]Product Rule Application: Using the product rule and the chain rule, we differentiate the function: d/dP [(379)/(4760)P(1 - (P)/(758))] = (379)/(4760)(1 - (P)/(758) - P/(758))Derivative Simplification: Simplify the derivative: d/dP [(dP)/(dt)] = (379)/(4760)(1 - 2P/(758))Critical Points Calculation: Set the derivative equal to zero to find the critical points: (379)/(4760)(1 - 2P/(758)) = 0Critical Point Verification: Solve for P: 1 - 2P/(758) = 0 2P/(758) = 1 P = 758/2 P = 379We found that the critical point is at P = 379. To confirm that this is a maximum, we can use the second derivative test or analyze the behavior of the derivative around this point. However, since the rate of change (dP)/(dt) is a product of P and a linear function of P, and given the context of the problem, it is reasonable to conclude that P = 379 is where P(t) is growing the fastest without further testing.

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