The population of a mining city grows at a rate proportional to that population, in two years the population has doubled and a year later there were 10,000 inhabitants.
what was the initial population?

The initial population was approximatedly 3535 inhabitants.

Step-by-step explanation:

The population of the city can be given by the following differential equation.

,

In which r is the rate of growth of the population.

We can solve this diffential equation by the variable separation method.

Integrating both sides:

Since ln and the exponential are inverse operations, to write P in function of t, we apply ln to both sides.

C is the initial population, so:

Now, we apply the problem's statements to first find the growth rate and then the initial population.

The problem states that:

In two years the population has doubled:

To isolate r, we apply ln both sides

So

In two years the population has doubled and a year later there were 10,000 inhabitants.

The initial population was approximatedly 3535 inhabitants.