The rate of change (dP)/(dt) of the number of fox on an island is modeled by the following differential equation: (dP)/(dt)=(1386)/(3791)P(1-(P)/( 616)) At t=0, the number of fox on the island is 170 and is increasing at a rate of 45 fox per day. At what value of P is P(t) growing the fastest? Answer:

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The rate of change  (dP)/(dt) of the number of fox on an island is modeled by the following differential equation:  (dP)/(dt)=(1386)/(3791)P(1-(P)/( 616)) At  t=0, the number of fox on the island is 170 and is increasing at a rate of 45 fox per day. 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) = (1386)/(3791)P(1 - (P)/(616)) 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. This is because at the maximum growth rate, the slope of the tangent to the curve representing (dP)/(dt) is horizontal, which corresponds to a derivative of zero.Find Derivative of dP/dt: Let's find the derivative of (dP)/(dt) with respect to P: (d^2P)/(dt^2) = d/dP[(1386)/(3791)P(1 - (P)/(616))] To simplify the differentiation, let's first simplify the expression inside the derivative: (d^2P)/(dt^2) = d/dP[(1386)/(3791)P - (1386)/(3791)(P^2)/(616)]Simplify Differentiation: Now, differentiate the expression with respect to P: (d^2P)/(dt^2) = (1386)/(3791) - (1386)/(3791)(2P)/(616) Simplify the coefficients: (d^2P)/(dt^2) = (1386)/(3791) - (1386*2)/(3791*616)PSet Second Derivative Equal to Zero: Set the second derivative equal to zero to find the critical points: 0 = (1386)/(3791) - (1386*2)/(3791*616)P Multiply through by (3791*616) to clear the denominators: 0 = (1386*616) - (1386*2)PSolve for P: Solve for P: (1386*2)P = (1386*616) P = (1386*616)/(1386*2) P = 616/2 P = 308Verify Maximum Point: We found that the value of P where P(t) is growing the fastest is 308. However, we should verify that this is indeed a maximum by checking the second derivative's sign around the critical point. If the second derivative is negative before and after P = 308, then P = 308 is a maximum.Check Second Derivative at P = 307: Substitute a value less than 308 into the second derivative and check the sign: (d^2P)/(dt^2) at P = 307 = (1386)/(3791) - (1386*2)/(3791*616)*307 This should yield a positive value, indicating that the curve is concave up before P = 308.Check Second Derivative at P = 309: Substitute a value greater than 308 into the second derivative and check the sign: (d^2P)/(dt^2) at P = 309 = (1386)/(3791) - (1386*2)/(3791*616)*309 This should yield a negative value, indicating that the curve is concave down after P = 308.Confirm Maximum Point: Since the second derivative changes from positive to negative as P increases through 308, this confirms that P = 308 is indeed the value where P(t) is growing the fastest.

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