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The rate of change (dP)/(dt) of the number of people on an island is modeled by the following differential equation: (dP)/(dt)=(17753)/(53132)P(1-(P)/( 866)) At t=0, the number of people on the island is 148 and is increasing at a rate of 41 people per hour. At what value of P is P(t) growing the fastest? Answer:

The rate of change dPdt \frac{d P}{d t} of the number of people on an island is modeled by the following differential equation:\newlinedPdt=1775353132P(1−P866) \frac{d P}{d t}=\frac{17753}{53132} P\left(1-\frac{P}{866}\right) \newlineAt t=0 t=0 , the number of people on the island is 148148 and is increasing at a rate of 4141 people per hour. At what value of P P is P(t) P(t) growing the fastest?\newlineAnswer:

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Q. The rate of change dPdt \frac{d P}{d t} of the number of people on an island is modeled by the following differential equation:\newlinedPdt=1775353132P(1−P866) \frac{d P}{d t}=\frac{17753}{53132} P\left(1-\frac{P}{866}\right) \newlineAt t=0 t=0 , the number of people on the island is 148148 and is increasing at a rate of 4141 people per hour. At what value of P P is P(t) P(t) growing the fastest?\newlineAnswer:
  1. Given Differential Equation: The given differential equation is:\newline(dPdt)=1775353132⋅P⋅(1−P866)(\frac{dP}{dt}) = \frac{17753}{53132} \cdot P \cdot (1 - \frac{P}{866})\newlineTo find the value of PP where P(t)P(t) is growing the fastest, we need to find the maximum point of the rate of change function, (dPdt)(\frac{dP}{dt}). This occurs when the derivative of (dPdt)(\frac{dP}{dt}) with respect to PP is equal to zero.
  2. Derivative Setup: Let's set up the derivative of dPdt\frac{dP}{dt} with respect to PP:ddP[dPdt]=ddP[1775353132⋅P⋅(1−P866)]\frac{d}{dP} \left[\frac{dP}{dt}\right] = \frac{d}{dP} \left[\frac{17753}{53132} \cdot P \cdot \left(1 - \frac{P}{866}\right)\right]We apply the product rule and the chain rule to find this derivative.
  3. Derivative Calculation: The derivative is:\newlineddP(1775353132⋅P⋅(1−P866))=1775353132⋅(1−P866)+1775353132⋅P⋅(−1866)\frac{d}{dP} \left(\frac{17753}{53132} \cdot P \cdot \left(1 - \frac{P}{866}\right)\right) = \frac{17753}{53132} \cdot \left(1 - \frac{P}{866}\right) + \frac{17753}{53132} \cdot P \cdot \left(-\frac{1}{866}\right)\newlineSimplify the expression.
  4. Simplify Expression: After simplifying, we get:\newline1775353132−2×17753×P53132×866\frac{17753}{53132} - \frac{2 \times 17753 \times P}{53132 \times 866}\newlineNow, we set this derivative equal to zero to find the critical points.
  5. Critical Points: Setting the derivative equal to zero gives us:\newline rac{17753}{53132} - rac{2 \times 17753 \times P}{53132 \times 866} = 0\newlineMultiply through by 53132×86653132 \times 866 to clear the denominators.
  6. Solve for P: We obtain: \newline17753×866−2×17753×P=017753 \times 866 - 2 \times 17753 \times P = 0\newlineSolve for P.
  7. Isolate P: Divide both sides by (2×17753)(2 \times 17753) to isolate P:\newlineP=(17753×866)(2×17753)P = \frac{(17753 \times 866)}{(2 \times 17753)}
  8. Calculate Value: Simplify the expression: P=8662P = \frac{866}{2}
  9. Calculate Value: Simplify the expression:\newlineP=8662P = \frac{866}{2}Calculate the value of PP:\newlineP=433P = 433\newlineThis is the value of PP where P(t)P(t) is growing the fastest.

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