## Sunday, April 21, 2013

### Exponential Growth - Population Problem

Category: Algebra

"Published in Suisun City, California, USA"

The population of California was 10,586,223 in 1950 and 23,668,562 in 1980. Assume the population grows exponentially.

(a) Find a formula for the population t years after 1950.
(b) Find the time required for the population to double.
(c) Use the data to predict the present population of California.

Solution:

The given problem above is about the exponential growth for a population in California. The exponential growth is given by the formula

where

x = population at time t
x0 = initial size of population
r = relative rate of growth (expressed as a proportion of the population)
t = time of growth

Now, if the population of California in 1950 (initial time) is 10,586,223 and in 1980 (final time) is 23,668,562, the growth rate r will be equal to

Take natural logarithm on both sides of the equation, we have

Therefore, the population of California in time t is

After 1950, the population of California will be doubled for

Take natural logarithm on both sides of the equation, we have

or

Therefore, the population of California will be doubled in 1950 + 26 = 1976.

In 2013, the population of California is

or