To determine when the population of the country will be 275 million using the exponential model [tex]\( A = 108.7 e^{0.032 t} \)[/tex], follow these steps:
1. Understand the given equation:
[tex]\[
A = 108.7 e^{0.032 t}
\][/tex]
where [tex]\( A \)[/tex] is the population in millions, and [tex]\( t \)[/tex] is the number of years after 2003.
2. Set up the equation with the target population:
We need to find [tex]\( t \)[/tex] such that the population [tex]\( A \)[/tex] is 275 million.
[tex]\[
275 = 108.7 e^{0.032 t}
\][/tex]
3. Solve for [tex]\( e^{0.032 t} \)[/tex]:
Divide both sides of the equation by 108.7:
[tex]\[
e^{0.032 t} = \frac{275}{108.7}
\][/tex]
Calculate the division:
[tex]\[
e^{0.032 t} \approx 2.5298
\][/tex]
4. Isolate the exponent [tex]\( 0.032 t \)[/tex] using natural logarithms:
Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(e^{0.032 t}) = \ln(2.5298)
\][/tex]
5. Simplify using the property of logarithms ([tex]\(\ln(e^x) = x\)[/tex]):
[tex]\[
0.032 t = \ln(2.5298)
\][/tex]
6. Solve for [tex]\( t \)[/tex]:
Compute the value of [tex]\(\ln(2.5298)\)[/tex]:
[tex]\[
0.032 t \approx 0.9275
\][/tex]
Divide both sides by 0.032:
[tex]\[
t \approx \frac{0.9275}{0.032} \approx 29.0056
\][/tex]
7. Round to the nearest year:
[tex]\[
t \approx 29
\][/tex]
So, the population of the country will be 275 million in approximately 29 years after 2003. To find the exact year:
[tex]\[
2003 + 29 = 2032
\][/tex]
Therefore, the population of the country will be 275 million in 2032.
