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------------------------------------------------ A sociologist is studying the population growth of a city. The city's current population is 71,000, and the sociologist estimates that the population will increase by [tex]5\%[/tex] every year.

The sociologist wants to find the number of years, [tex]t[/tex], it will take for the city's population to reach at least 85,000. Which inequality can she use?

A. [tex]71,000(0.95)^t \geq 85,000[/tex]

B. [tex]71,000 t^5 \geq 85,000[/tex]

C. [tex]71,000(1.5)^t \geq 85,000[/tex]

D. [tex]71,000(1.05)^t \geq 85,000[/tex]

E. [tex]81,000 t \geq 71,000^{1.05}[/tex]

F. [tex]14,000(1.05)^t \geq 85,000[/tex]

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]