Question

What is the exponential equation that models the population of the city Laredo?Laredo that had a population of 12,600 in 2000.By the year 2006 the population grew to 13,542.(write your equation for p in terms of n . round your b value to 4 decimal places)Use your equation to predict the population in the year 2015The population of Laredo in 2015 should beIn what year will the population reach 28,000 ?

Answer 1

Exponential growth formula

$p=a\cdot b^n$

where

• p: population in year n

,

• a: initial population

,

• b: growth factor (must be greater than 1)

,

• n: time, in years

Taking the year 2000 as n = 0, then initial population is a = 12,600.

In year 2006, n = 6 and p = 13,542. Substituting these values and solving for b:

$\begin{gathered} 13542=12600\cdot b^6 \\ \frac{13542}{12600}=b^6 \\ \log _{10}(\frac{13542}{12600})=6\cdot\log _{10}b \\ \frac{\log _{10}(\frac{13542}{12600})}{6}=\log _{10}b \\ 10^{0.0052}\approx b \\ 1.0121\approx b \end{gathered}$

In year 2015, n = 15, therefore

$\begin{gathered} p=12600\cdot1.0121^{15} \\ p\approx15091 \end{gathered}$

The population of Laredo in 2015 should be approximately 15,091

Substituting p = 28,000 into the equation and solving for n:

$\begin{gathered} 28000=12600\cdot1.0121^n \\ \frac{28000}{12600}=1.0121^n \\ \log _{10}(\frac{28000}{12600})=n\cdot\log _{10}1.0121 \\ \frac{\log _{10}(\frac{28000}{12600})}{\log _{10}1.0121}=n \\ 67\approx n \end{gathered}$

The population will reach 28,000 in 2067 (=2000+67)