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Answer :
To determine which equation represents the town's population after 6 years, we need to use the formula for exponential growth. When a population grows at a fixed percentage rate per year, we can use the following equation:
[tex]\[ P = P_0 (1 + r)^t \][/tex]
where:
- [tex]\( P \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (expressed as a decimal), and
- [tex]\( t \)[/tex] is the time in years.
Given:
- Initial population, [tex]\( P_0 = 141,300 \)[/tex]
- Growth rate, [tex]\( r = 5\% = 0.05 \)[/tex] (as a decimal)
- Time, [tex]\( t = 6 \)[/tex] years
Now, plug these values into the formula:
[tex]\[ P = 141,300 \times (1 + 0.05)^6 \][/tex]
This equation accounts for the compounding effect of the population growing by 5% every year for 6 years.
Therefore, the equation that represents the town's population after 6 years is:
[tex]\[ P = 141,300(1+0.05)^6 \][/tex]
This equation reflects the population growth using a 5% increase compounded annually for 6 years, and it is the correct choice from the options provided.
[tex]\[ P = P_0 (1 + r)^t \][/tex]
where:
- [tex]\( P \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (expressed as a decimal), and
- [tex]\( t \)[/tex] is the time in years.
Given:
- Initial population, [tex]\( P_0 = 141,300 \)[/tex]
- Growth rate, [tex]\( r = 5\% = 0.05 \)[/tex] (as a decimal)
- Time, [tex]\( t = 6 \)[/tex] years
Now, plug these values into the formula:
[tex]\[ P = 141,300 \times (1 + 0.05)^6 \][/tex]
This equation accounts for the compounding effect of the population growing by 5% every year for 6 years.
Therefore, the equation that represents the town's population after 6 years is:
[tex]\[ P = 141,300(1+0.05)^6 \][/tex]
This equation reflects the population growth using a 5% increase compounded annually for 6 years, and it is the correct choice from the options provided.
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