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A population of fruit flies in a laboratory experiment is modelled by the logistic function:$$ P(t) = \frac{L}{1 + Ce^{-kt}} $$where \(t\) is the time in days.
The carrying capacity of the environment is known to be 500 flies.
Initially (\(t=0\)), there were 50 flies.
After 2 days (\(t=2\)), the population grew to 150 flies.
State the value of \(L\).
Use the initial population to find the value of \(C\).
Use the population at \(t=2\) to find the value of \(k\).
Write the complete equation for \(P(t)\).
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