Assuming that P ? 0, a population is modeled by the differential equation 1. For
ID: 2866440 • Letter: A
Question
Assuming that P ? 0, a population is modeled by the differential equation
1. For what values of P is the population increasing? Answer (in interval notation):
2. For what values of P is the population decreasing? Answer (in interval notation):
3. What are the equilibrium solutions? Answer (separate by commas): P =
Assuming that P ? 0, a population is modeled by the differential equation dP/dt = 1.1P(1- P/4100) 1. For what values of P is the population increasing? Answer (in interval notation): 2. For what values of P is the population decreasing? Answer (in interval notation): 3. What are the equilibrium solutions? Answer (separate by commas): P =Explanation / Answer
A. Population is increasing when dP/dt > 0
1.1 P (1 - P/4100) > 0
Now population is never negative, so we divide both sides by 1.1 P (P > 0)
1 - P/4100 > 0
1 > P/4100
4100 > P
P < 4100
Since P > 0, then we get 0 < P < 4100
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B. Population is decreasing when dP/dt < 0
1.1 P (1 - P/4100) < 0
Now population is never negative, so we divide both sides by 1.1 P (P > 0)
1 - P/4100 < 0
1 < P/4100
4100 < P
P > 4100
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C. Equilibrium occurs when dP/dt = 0
1.1P (1 - P/4100) = 0
P = 0 or (1 - P/4100) = 0
P = 0 or 4100
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Explanation:
If original population = 0, then it cannot grow and stays at 0 (equilibrium)
If original population is between 0 and 4100, then population increases, until it reaches 4100, at which point it reaches equilibrium (remains constant at 4100)
If original population is above 4100, then population decreases, until it reaches 4100, at which point it reaches equilibrium