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Answer :
To solve this question, we need to determine which equation correctly represents the population of the town after 8 years, given that the population shrinks by 6% each year.
1. Understand the Problem:
- Initial population = 141,200
- Annual shrinkage rate = 6% or 0.06
- Number of years = 8
2. Set Up the Equation:
- Since the population decreases by 6% each year, the remaining population after each year is 94% of the previous year's population. This can be calculated as (1 - 0.06), or 0.94.
- To find the population after 8 years, we raise 0.94 to the 8th power (because the population shrinks by 6% each year for 8 years).
3. Write the Correct Exponential Decay Equation:
- The equation should be:
[tex]\[
P = 141,200 \times (1 - 0.06)^8
\][/tex]
- This equation accurately represents the situation by considering the shrinkage rate over the 8-year period.
4. Choose the Correct Option:
- Among the options given, the correct equation is:
[tex]\[
P = 141,200 \times (1 - 0.06)^8
\][/tex]
This explains why the option [tex]\( P = 141,200(1-0.06)^8 \)[/tex] is the correct choice, showing how the population decreases over 8 years due to the yearly shrinkage rate of 6%.
1. Understand the Problem:
- Initial population = 141,200
- Annual shrinkage rate = 6% or 0.06
- Number of years = 8
2. Set Up the Equation:
- Since the population decreases by 6% each year, the remaining population after each year is 94% of the previous year's population. This can be calculated as (1 - 0.06), or 0.94.
- To find the population after 8 years, we raise 0.94 to the 8th power (because the population shrinks by 6% each year for 8 years).
3. Write the Correct Exponential Decay Equation:
- The equation should be:
[tex]\[
P = 141,200 \times (1 - 0.06)^8
\][/tex]
- This equation accurately represents the situation by considering the shrinkage rate over the 8-year period.
4. Choose the Correct Option:
- Among the options given, the correct equation is:
[tex]\[
P = 141,200 \times (1 - 0.06)^8
\][/tex]
This explains why the option [tex]\( P = 141,200(1-0.06)^8 \)[/tex] is the correct choice, showing how the population decreases over 8 years due to the yearly shrinkage rate of 6%.
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