Answer :
Alright, let's walk through the problem step-by-step to estimate the population in 2003 using the exponential growth formula.
### Step 1: Understand the Given Data
- Population in 1991 ([tex]\( P_{1991} \)[/tex]): 231 million
- Population in 1999 ([tex]\( P_{1999} \)[/tex]): 233 million
- We need to estimate the population in 2003.
- The time difference from 1991 to 1999 is [tex]\( t = 1999 - 1991 = 8 \)[/tex] years.
### Step 2: The Exponential Growth Formula
The exponential growth formula is given by:
[tex]\[ P = A e^{kt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( A \)[/tex] is the initial population,
- [tex]\( e \)[/tex] is the base of the natural logarithm,
- [tex]\( k \)[/tex] is the growth rate,
- [tex]\( t \)[/tex] is the time in years since the initial population was recorded.
### Step 3: Estimate the Growth Rate [tex]\( k \)[/tex]
To estimate the growth rate [tex]\( k \)[/tex], we use the known populations in 1991 and 1999:
[tex]\[ P_{1999} = P_{1991} e^{k \cdot (1999-1991)} \][/tex]
[tex]\[ 233 = 231 e^{k \cdot 8} \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ \frac{233}{231} = e^{8k} \][/tex]
[tex]\[ \ln \left( \frac{233}{231} \right) = 8k \][/tex]
[tex]\[ k = \frac{\ln \left( \frac{233}{231} \right)}{8} \][/tex]
Using a calculator, we find:
[tex]\[ k \approx 0.0010776 \][/tex]
(rounded to four decimal places, [tex]\( k \approx 0.0011 \)[/tex]).
### Step 4: Calculate the Population in 2003
Now, we need to estimate the population in 2003. The time difference from 1991 to 2003 is [tex]\( t = 2003 - 1991 = 12 \)[/tex] years.
Using the exponential growth formula:
[tex]\[ P_{2003} = P_{1991} e^{kt} \][/tex]
[tex]\[ P_{2003} = 231 \times e^{0.0010776 \times 12} \][/tex]
Using a calculator, we find:
[tex]\[ P_{2003} \approx 234.00648416664575 \][/tex]
### Step 5: Round the Population to the Nearest Million
Finally, we round the calculated population to the nearest million:
[tex]\[ P_{2003} \approx 234 \text{ million} \][/tex]
### Conclusion
The estimated population in 2003 is 234 million.
Therefore, the correct answer is 234 million.
### Step 1: Understand the Given Data
- Population in 1991 ([tex]\( P_{1991} \)[/tex]): 231 million
- Population in 1999 ([tex]\( P_{1999} \)[/tex]): 233 million
- We need to estimate the population in 2003.
- The time difference from 1991 to 1999 is [tex]\( t = 1999 - 1991 = 8 \)[/tex] years.
### Step 2: The Exponential Growth Formula
The exponential growth formula is given by:
[tex]\[ P = A e^{kt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( A \)[/tex] is the initial population,
- [tex]\( e \)[/tex] is the base of the natural logarithm,
- [tex]\( k \)[/tex] is the growth rate,
- [tex]\( t \)[/tex] is the time in years since the initial population was recorded.
### Step 3: Estimate the Growth Rate [tex]\( k \)[/tex]
To estimate the growth rate [tex]\( k \)[/tex], we use the known populations in 1991 and 1999:
[tex]\[ P_{1999} = P_{1991} e^{k \cdot (1999-1991)} \][/tex]
[tex]\[ 233 = 231 e^{k \cdot 8} \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ \frac{233}{231} = e^{8k} \][/tex]
[tex]\[ \ln \left( \frac{233}{231} \right) = 8k \][/tex]
[tex]\[ k = \frac{\ln \left( \frac{233}{231} \right)}{8} \][/tex]
Using a calculator, we find:
[tex]\[ k \approx 0.0010776 \][/tex]
(rounded to four decimal places, [tex]\( k \approx 0.0011 \)[/tex]).
### Step 4: Calculate the Population in 2003
Now, we need to estimate the population in 2003. The time difference from 1991 to 2003 is [tex]\( t = 2003 - 1991 = 12 \)[/tex] years.
Using the exponential growth formula:
[tex]\[ P_{2003} = P_{1991} e^{kt} \][/tex]
[tex]\[ P_{2003} = 231 \times e^{0.0010776 \times 12} \][/tex]
Using a calculator, we find:
[tex]\[ P_{2003} \approx 234.00648416664575 \][/tex]
### Step 5: Round the Population to the Nearest Million
Finally, we round the calculated population to the nearest million:
[tex]\[ P_{2003} \approx 234 \text{ million} \][/tex]
### Conclusion
The estimated population in 2003 is 234 million.
Therefore, the correct answer is 234 million.
