# Calculus

posted by Brian J

Assuming the population of the earth changes at a rate proportional to the current population further, it is estimated that at time t=0, the earth's population was 600 million, at t=300, it's population was 2.8 billion. find an expression giving the population of the earth at anytime. assuming that the greatest population the earth can support is 25 billion, when will this limit be reached?

1. Steve

population changes at a rate proportional to the current population:
dp/dt = kp

dp/p = k dt
ln p = kt+c
p = c*e^(kt)
Let p be measured in units of millions. Then we have
p(0) = 600
p(300) = 2800
so,
p(t) = 600e^(kt)
600e^(300k) = 2800
e^(300k) = 14/3
300k = ln(14/3)
k = ln(14/3)/300 = 0.00513

p(t) = 600 e^(0.00513t)

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