To formulate the equation that gives the population [tex]\( A(z) \)[/tex] of the town [tex]\( z \)[/tex] years since 2005, follow these steps:
1. Identify the initial population: The initial population, which we denote as [tex]\( P_0 \)[/tex], is given as 12,500.
2. Determine the annual growth rate: The annual growth rate is 5.4%. To use it in our calculations, we need to convert this percentage into a decimal. This conversion is done by dividing the percentage by 100:
[tex]\[
\text{Annual growth rate as a decimal} = \frac{5.4}{100} = 0.054
\][/tex]
3. Calculate the growth multiplier: The population grows by 5.4% each year, which means each year it is multiplied by 1.054 (100% + 5.4%):
[tex]\[
\text{Growth multiplier} = 1 + 0.054 = 1.054
\][/tex]
4. Formulate the equation: The population after [tex]\( z \)[/tex] years is given by the initial population multiplied by the growth multiplier raised to the power of [tex]\( z \)[/tex]. This relationship can be expressed as:
[tex]\[
A(z) = P_0 \cdot (\text{Growth multiplier})^z
\][/tex]
Substituting the values we have identified:
[tex]\[
A(z) = 12500 \cdot (1.054)^z
\][/tex]
Therefore, the equation that gives the population [tex]\( A(z) \)[/tex] of the town [tex]\( z \)[/tex] years since 2005 is:
[tex]\[
A(z) = 12500 \cdot (1.054)^z
\][/tex]
