Answer :
To find the number of years, [tex]\( t \)[/tex], it will take for the city's population to reach at least 85,000, we need to set up an inequality that describes the situation.
1. Current Situation:
The city's current population is 71,000.
2. Annual Increase:
The population increases by 5% each year. To find the population after each year, we multiply the current population by 1 plus the growth rate (which is 0.05 for 5%).
So, each year, the population becomes 71,000 multiplied by [tex]\( (1 + 0.05) = 1.05 \)[/tex].
3. Expression for Future Population:
After [tex]\( t \)[/tex] years, the population will be [tex]\( 71,000 \times (1.05)^t \)[/tex].
4. Setting Up the Inequality:
We want to find when this expression is at least 85,000. So, we set up the inequality:
[tex]\[
71,000 \times (1.05)^t \geq 85,000
\][/tex]
This inequality will allow the sociologist to solve for [tex]\( t \)[/tex] to find the number of years it will take for the population to be at least 85,000. Therefore, the correct inequality to use is:
[tex]\[
71,000 (1.05)^t \geq 85,000
\][/tex]
1. Current Situation:
The city's current population is 71,000.
2. Annual Increase:
The population increases by 5% each year. To find the population after each year, we multiply the current population by 1 plus the growth rate (which is 0.05 for 5%).
So, each year, the population becomes 71,000 multiplied by [tex]\( (1 + 0.05) = 1.05 \)[/tex].
3. Expression for Future Population:
After [tex]\( t \)[/tex] years, the population will be [tex]\( 71,000 \times (1.05)^t \)[/tex].
4. Setting Up the Inequality:
We want to find when this expression is at least 85,000. So, we set up the inequality:
[tex]\[
71,000 \times (1.05)^t \geq 85,000
\][/tex]
This inequality will allow the sociologist to solve for [tex]\( t \)[/tex] to find the number of years it will take for the population to be at least 85,000. Therefore, the correct inequality to use is:
[tex]\[
71,000 (1.05)^t \geq 85,000
\][/tex]
