The number of fish in a lake is decreasing. After \(t\) years, there are \(x\) fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
Formulate a differential equation, in the variables \(x\) and \(t\) and a constant of proportionality \(k\), where \(k > 0\), to model the rate at which the number of fish in the lake is decreasing.
At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of \(k\).
The equation
$$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$
is proposed as a model for the number of fish, \(P\), in another lake, where \(t\) is the time in years and \(A\) is a positive constant.
On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
Taking 1 January 2008 as \(t = 0\), find the value of \(A\).
Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.