To find the number of years it would take for the population to exceed 9,000, we should follow these steps:
1. Set the population equation greater than 9,000:
[tex]\[
2400 \cdot (1 + 0.02)^t > 9000
\][/tex]
2. Take the logarithm of both sides of the inequality to help solve for [tex]\( t \)[/tex]:
[tex]\[
\log(2400 \cdot (1.02)^t) > \log(9000)
\][/tex]
3. Apply the properties of logarithms to simplify:
[tex]\[
\log(2400) + \log((1.02)^t) > \log(9000)
\][/tex]
Which can further be simplified using the power rule of logarithms:
[tex]\[
\log(2400) + t \cdot \log(1.02) > \log(9000)
\][/tex]
4. Isolate [tex]\( t \)[/tex]:
[tex]\[
t \cdot \log(1.02) > \log(9000) - \log(2400)
\][/tex]
[tex]\[
t > \frac{\log(9000) - \log(2400)}{\log(1.02)}
\][/tex]
So, the correct steps include:
- Set the population greater than 9000 (Option c).
- Take the log of both sides (Option d).
Therefore, the selected options should be:
- c. Set the population greater than 9000
- d. Take log of both sides
To summarize, the inequality that isolates [tex]\( t \)[/tex] is:
[tex]\[
t > \frac{\log(9000) - \log(2400)}{\log(1.02)}
\][/tex]
