Applied Mathematics

A differential equation describes the rate of change of a population of bacteria in a Petri dish:

(frac{{dP}}{{dt}} = kP(1 – frac{{P}}{{N}}))

Where:

– (frac{{dP}}{{dt}}) represents the rate of change of the population with respect to time.

– (P) is the population of bacteria at time (t).

– (N) is the carrying capacity of the Petri dish (the maximum population it can support).

– (k) is a constant representing the growth rate.

Suppose (N = 500) and (k = 0.02). Find the solution to this differential equation with an initial condition (P(0) = 10), which means there were initially 10 bacteria in the Petri dish at time (t = 0). What will be the population of bacteria at (t = 10)?

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