The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P0 has doubled in 5 years.1) Suppose it is known that the population is 9000 after 3 years. What was the initial population P0?2) What will the population be in 10 years?3) How fast is the population growing at t =10?

1)

For this case we know that the population is doubled after 5 years so then we can assume that the initial population is Po and we have this:

We can apply natural log on both sides and we got:

We know also that the population after 3 years is 9000, so we can use this condition:

And we can solve for Po and we got:

2)

If we replace the value of t =10 we got:

3)  

Since we know the value of the population at t=10 , P= 23751.144 and k = 0.138629436 we can replace and we got:

Explanation:

Part 1

For this case we know that we can use the proportional model given by:

We can rewrite the expression like this:

If we integrate both sides we got:

After apply exponentials in both sides we got:

For this case we know that the population is doubled after 5 years so then we can assume that the initial population is Po and we have this:

We can apply natural log on both sides and we got:

We know also that the population after 3 years is 9000, so we can use this condition:

And we can solve for Po and we got:

Part 2

For this case we have the following model:

If we replace the value of t =10 we got:

Part 3

For this case we know that:

Since we know the value of the population at t=10 , P= 23751.144 and k = 0.138629436 we can replace and we got: