To determine which equations correctly represent the population of Smalltown [tex]\( t \)[/tex] years after 2010 given the exponential growth rate, we need to evaluate the options provided based on our knowledge of exponential growth formulas.
The initial population in 2010 is 56,382 people.
Exponential growth can be described by two main forms of equations:
1. [tex]\( P = P_0 \left(1 + r\right)^t \)[/tex]
2. [tex]\( P = P_0 e^{rt} \)[/tex]
Where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the number of years after the starting point
Given the information:
- The initial population [tex]\( P_0 \)[/tex] is 56,382.
- The annual growth rate [tex]\( r \)[/tex] is 1.2 percent, or 0.012 in decimal form.
Let's evaluate each provided equation:
1. [tex]\( P=56,382(1+0.012)^t \)[/tex]:
- This equation uses the first form: [tex]\( P = P_0 \left(1 + r\right)^t \)[/tex].
- Here, [tex]\( P_0 = 56,382 \)[/tex], [tex]\( r = 0.012 \)[/tex], and the form matches exactly.
- This equation is correct.
2. [tex]\( P=56,382 e^{0.012 t} \)[/tex]:
- This equation uses the second form: [tex]\( P = P_0 e^{rt} \)[/tex].
- Here, [tex]\( P_0 = 56,382 \)[/tex], [tex]\( r = 0.012 \)[/tex], and the form matches exactly.
- This equation is correct.
3. [tex]\( P=56,382(1+0.12)^t \)[/tex]:
- This equation also uses the first form but incorrectly represents the growth rate as 0.12 instead of 0.012.
- Thus, this equation is incorrect.
4. [tex]\( P=56,382 e^{1.2 t} \)[/tex]:
- This equation uses the second form but incorrectly represents the growth rate as 1.2 instead of 0.012.
- Thus, this equation is incorrect.
From the evaluation, we can determine:
- The correct equations are [tex]\( P=56,382(1+0.012)^t \)[/tex] and [tex]\( P=56,382 e^{0.012 t} \)[/tex].
Therefore, the correct options are:
[tex]\( \boxed{1 \text{ and } 2} \)[/tex]
