The population of a suburb grows at a rate proportional to the population. suppose the population doubles in size from 3000 to 6000 in a 6- month period and continues at the current rate of growth. find the particular solution to the differential equation with the initial condition p(0)=3000.

Step-by-step explanation:

The growth of the population can be modeled by the following differential equation:

Where r is the growth rate, P is the population, and t is the time measures in months.

I am going to solve the above differential equation with the separation of variables method.

Integrating both sides:

Where P(0) is the initial condition

We need to isolate P in this equation, so we do this

So

The problem states that P(0)=3000, so:

The problem wants us to find the value of r:

It states that the population doubles in size from 3000 to 6000 in a 6- month period, meaning that P(6) = 6000. So

To isolate 6r, we apply ln to both sides.

r = 0.1155

The particular solution to the differential equation with the initial condition P(0)=3000 is:

the answer is 99.