To determine the population of the country in 2003 using the exponential model [tex]\( A = 815.4 e^{0.002t} \)[/tex]:
1. Identify the key parameters in the model:
- [tex]\( A \)[/tex] is the population of the country in millions.
- [tex]\( t \)[/tex] is the number of years after 2003.
- The base population at [tex]\( t = 0 \)[/tex] is 815.4 million.
- The growth rate is 0.002 per year.
2. To find the population in 2003, set [tex]\( t \)[/tex] to 0 since 2003 is the base year from which time is measured.
3. Substitute [tex]\( t = 0 \)[/tex] into the exponential model:
[tex]\[
A = 815.4 e^{0.002 \times 0}
\][/tex]
4. Simplify the exponent:
[tex]\[
A = 815.4 e^0
\][/tex]
5. Recall that [tex]\( e^0 = 1 \)[/tex]:
[tex]\[
A = 815.4 \times 1
\][/tex]
6. Therefore:
[tex]\[
A = 815.4
\][/tex]
The population of the country in 2003 was [tex]\( 815.4 \)[/tex] million.
