Respuesta :
The situation describes an exponential growth, which can be expressed using the general formula:
[tex]y=a(1+r)^x[/tex]Where
a is the initial value
r is the growth rate, expressed as a decimal value
x is the number of times intervals
y is the final value after x time intervals
For the studied population, the growth rate is 2.5%, to express this number as a decimal value you have to divide it by 100:
[tex]\begin{gathered} r=\frac{2.5}{100} \\ r=0.025 \end{gathered}[/tex]The initial value is the current population of the town: a=50000
You can express the equation of exponential growth for this population as follows:
[tex]\begin{gathered} y=50000(1+0.025)^x \\ y=50000(1.025)^x \end{gathered}[/tex]We know that after x years the population will be y=100000, to determine how many years it will take to reach this value you have to equal the equation to 100000 and solve for x:
[tex]100000=50000(1.025)^x[/tex]-Divide both sides of the expression by 50000
[tex]\begin{gathered} \frac{100000}{50000}=\frac{50000(1.025)^x}{50000} \\ 2=(1.025)^x \end{gathered}[/tex]-Apply logarithm to both sides of the equal sign:
[tex]\begin{gathered} \log (2)=\log (1.025^x) \\ \log (2)=x\cdot\log (1.025) \end{gathered}[/tex]-Divide both sides by the logarithm of 1.025
[tex]\begin{gathered} \frac{\log(2)}{\log(1.025)}=\frac{x\cdot\log (1.025)}{\log (1.025)} \\ x\approx28.07\approx28 \end{gathered}[/tex]After approximately 28 years the population of the town will be 100,000.
