Exponential and Logarithmic Functions - Applications of Exponential and Logarithmic Functions

A certain species of fish was introduced into a pond and its number was tracked daily. It was observed that \(t\) days later, its population \(P\) was given by \(P=5000+A{{\mathrm{e}}^{kt}}\), where \(A\) and \(k\) are constants. Initially, there were \(3000\) fishes in the pond, and \(10\) days later, \(3800\) fishes were counted.

1. Explain why \(A=-2000\).
2. Determine the value of \(k\), giving your answer correct to \(3\) significant figures, where appropriate.
3. Find the number of days needed for the fish to increase the population by \(40\%\).
4. How many fishes are expected in the pond in the long run?