Sure! Let's break down the steps to determine the population of the country in the year 2003 using the given exponential model [tex]\( A = 222.6 e^{0.025t} \)[/tex].
1. Understand the model:
The given exponential model [tex]\( A = 222.6 e^{0.025t} \)[/tex] represents the population [tex]\( A \)[/tex] of a country (in millions) as a function of time [tex]\( t \)[/tex]. Here,
- 222.6 is the initial population (in millions) at the base year [tex]\( t = 0 \)[/tex].
- [tex]\( e \)[/tex] is the mathematical constant approximately equal to 2.71828.
- 0.025 is the growth rate per year.
- [tex]\( t \)[/tex] is the time elapsed in years from the base year.
2. Identify the base year and the year of interest:
To determine [tex]\( t \)[/tex], we need to know the base year. Suppose the base year is 2000. This means:
- At the base year 2000, [tex]\( t = 0 \)[/tex].
- We are interested in finding the population in 2003.
3. Calculate the time elapsed [tex]\( t \)[/tex]:
Since the base year is 2000 and the year of interest is 2003, the time elapsed [tex]\( t \)[/tex] is:
[tex]\[ t = 2003 - 2000 = 3 \][/tex]
4. Substitute [tex]\( t \)[/tex] into the exponential model:
Now we need to substitute [tex]\( t = 3 \)[/tex] into the model to find the population in 2003:
[tex]\[ A = 222.6 e^{0.025 \cdot 3} \][/tex]
5. Calculate the exponent:
First calculate the exponent part [tex]\(0.025 \times 3\)[/tex]:
[tex]\[ 0.025 \times 3 = 0.075 \][/tex]
6. Evaluate the exponential term:
Next, evaluate [tex]\( e^{0.075} \)[/tex]:
[tex]\[ e^{0.075} \approx 1.077884 \][/tex] (using a calculator for precision)
7. Calculate the population by multiplying the initial population by the exponential term:
[tex]\[ A = 222.6 \times 1.077884 \][/tex]
[tex]\[ A \approx 239.937 \][/tex]
Therefore, the population of the country in 2003 was approximately [tex]\( 239.937 \)[/tex] million.
