The exponential model [tex]A = 108.7 e^{0.032 t}[/tex] describes the population [tex]\(A\)[/tex] of a country in millions, [tex]\(t\)[/tex] years after 2003.

Use the model to determine when the population of the country will be 275 million.

The population of the country will be 275 million in [tex]\(\square\)[/tex].

(Round to the nearest year as needed.)



Answer :

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.