Biologists stocked a lake with $300$ fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be $7000$. The number of fish doubled in the first year.

(a) Assuming that the size of the fish population satisfies the logistic equation $$\frac{dP}{dt} = kP \left( 1 - \frac{P}{K} \right) ,$$ determine the constant $k$, and then solve the equation to find an expression for the size of the population after $t$ years.
$k=$ ,
$P(t)=$ .

(b) How long will it take for the population to increase to $3500$ (half of the carrying capacity)?
It will take years.