To determine the population of Jamaica in 2010, given it was expected to grow exponentially by 1.1% each year since 1990, we follow these steps:
1. Initial Population and Growth Rate:
- Initial population in 1990: [tex]\(2,466,000\)[/tex]
- Annual growth rate: [tex]\(1.1\%\)[/tex]
2. Duration:
- Number of years between 1990 and 2010: [tex]\(2010 - 1990 = 20\)[/tex] years
3. Exponential Growth Formula:
- The exponential growth formula is [tex]\( P(t) = P_0 \times (1 + r)^t \)[/tex]
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the growth rate (per time period)
- [tex]\( t \)[/tex] is the number of time periods
4. Substitute and Calculate:
- [tex]\( P_0 = 2,466,000 \)[/tex]
- [tex]\( r = 0.011 \)[/tex] (since [tex]\(1.1\%\)[/tex] as a decimal is [tex]\(0.011\)[/tex])
- [tex]\( t = 20 \)[/tex]
Now we plug these values into the formula:
[tex]\[
P(20) = 2,466,000 \times (1 + 0.011)^{20}
\][/tex]
- First, we calculate [tex]\( (1 + 0.011) \)[/tex]:
[tex]\[
1 + 0.011 = 1.011
\][/tex]
- Then, we raise [tex]\( 1.011 \)[/tex] to the power of 20:
[tex]\[
1.011^{20} \approx 1.248092
\][/tex]
- Finally, multiply this result by the initial population:
[tex]\[
P(20) = 2,466,000 \times 1.248092 \approx 3,069,136
\][/tex]
So, the expected population of Jamaica in 2010 would be approximately [tex]\( 3,069,136 \)[/tex].
Therefore, the correct answer is:
[tex]\[ 3,069,136 \][/tex]
