Chapter 3.2, Problem 30E

Solution

To find: The time in which population of a city/place will be a given count.

The population of Vegas Nevada has become 1 million in past $\approx 12$ years, i.e., in 2012.

Given data: The population of Vegas Nevada was 478000 in 2000. The growth rate of population is 6.28% per year. The time required for population reached 1 million.

Method/Formula used:

Suppose $P_0$ be the initial $\left(\text{at}t=0\right)$ population of a city/place, then the population $P_t$ in at any time t with the growth rate r % per year is given by

  $P_t=P_0{\left(1+r\right)}^t$ … (1)

Calculations:

The initial population $P_0$ of population of Vegas, Nevada in the year $\left(\text{at}t=0\right)$ was 478000. The growth rate of the population is $6.28\%$ per year. Suppose in time t the population of Vegas, Nevada becomes one million, i.e., $P_t$ equals 1000000.

Substitute 478000 for $P_0$ , $\frac{6.20}{100}$ for r , and 1000000 for $P_t$ in result (1),

  $\begin{array}{c}1000000=478000{\left(1+\frac{6.28}{100}\right)}^t\\ 1000000=478000{\left(\frac{106.28}{100}\right)}^t\\ {\left(1.0628\right)}^t=\frac{1000000}{478000}\\ {\left(1.0628\right)}^t=2.0921\end{array}$

Taking ${\log }_e$ both sides,

  $\begin{array}{c}{\left(1.0628\right)}^t=2.0921\\ t{\log }_e\left(1.0628\right)={\log }_{e}2.0921\\ t=\frac{{\log }_{e}2.0921}{{\log }_{e}1.0628}\\ =12.12\end{array}$

Thus, the population of Jacksonville Florida has become 1 million in past $\approx 12$ years, i.e., in 2012.