** **
**Solutions to Problem 1.**

(a) Jan. 1, 2005 corresponds to t = 15. We have

P(15) = 3.6 ( ) .

Hence, P(15)=5.9444. The population in 2005 is going to be about 5,944,400 people.

(b) To answer (b), we have to solve for t the following exponential equation

9 = 3.6 ( ).

We divide both sides by 3.6 and obtain

= 2.5 .

We take the natural logarithm of both sides to get

, which gives = 27.4053.

The population will reach 9 million in the year 2017.

(c) To find the
*doubling time, *
that is the time needed for the population to double, we have to solve for t the equation

7.2 = 3.6 ( ).

We divide both sides by 3.6 and obtain

2 = .

Taking the logarithm of both sides and solving for t, and dividing both sides by ln(1.034) gives us

= 20.7313 years.

The population will double in approximately 21 years.

It follows from properties of exponential functions that the population will double every 21 years, regardless of the initial population or time. So the doubling time for a given exponential function is constant.