two pre-calc logistic growth problems?
The differential equation dP/dt=.0004P(1500-P) describes the growth of a population, P. Find the rate at which the population is growing when it is growing at its fastest.
Another logistic equation given is dP/dt =.04P(800-P). Find the size of the population when it's growing at its fastest.
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1) dP/dt=.0004P(1500-P)
dP/dt = 0.6P - 0.0004P^2
d^2P/dt^2 = 0.6 - 0.0008P = 0
0.0008P = 0.6
P = 750
dP/dt = 0.0004P(1500 - P)
dP dt = 0.0004(750)(1500 - 750) = 225
2) dP/dt =.04P(800-P)
dP/dt = 32P - 0.4P^2
d^2P/dt^2 = 32 - 0.8P = 0
0.8P = 32
P = 40