Expert: Josh - 6/28/2006

Question
If the world's population grew by 2% in 1998 and continued at that rate, how long would it take Earth's population to double?
-20yrs.
-25yrs.
-30yrs.
-35yrs.

Answer
Nicolette,

To answer this, we refer to the exponential model.

Let A be the population at time t=0 (which for our purpose coincides with new year day in 1998). Time, "t", is measured in years. Thus, when t=1, we interpret this as the start of year 1999.

The population at time "t" is given by
 P(t)=A*exp(kt), [#1]
where k is an exponential constant.

Step 1: Working out the value of "k"

The growth experienced in 1998 is 2%, from t=0 (the first day of 1998) to t=1 (practically, the last day of 1998).

Substituting t=1 into equation [#1], we get P(1)=A*exp(k)=1.02A. The last equality is due to the fact that the population increased by 2%.

Thus, cancelling the common factor "A", we get
exp(k)=1.02, taking the natural logarithm (i.e., inverse of the exponential function), we get k=ln(1.02). You can use a calculator to obtain an approximate numerical value.

Step 2: Finding the time taken to double the population (at the start of 1998).

We seek "t" such that population P(t)=2A.
Referring to the same equation in [#1], we solve for
2A=A*exp(kt)
2=exp(kt)
ln(2)=kt
t=ln(2)/k, where k=ln(1.02) was found earlier.

Cheers:)