Answer :
Sure, let's work through the problem step by step to estimate the population in 2008 using the exponential growth formula.
1. Understand and Define the Given Values:
- [tex]\( P_{1992} = 147 \)[/tex] million (population in 1992)
- [tex]\( P_{1996} = 153 \)[/tex] million (population in 1996)
- [tex]\( t_1 = 1996 - 1992 = 4 \)[/tex] years (time interval from 1992 to 1996)
- [tex]\( t_{2008} = 2008 - 1992 = 16 \)[/tex] years (time interval from 1992 to 2008)
2. Exponential Growth Formula:
The population growth can be modeled using the formula:
[tex]\[ P = A e^{k t} \][/tex]
where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( A \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the time
3. Calculate the Growth Rate [tex]\( k \)[/tex]:
To find [tex]\( k \)[/tex], we rearrange the formula to solve for [tex]\( k \)[/tex] using the known populations at two different times.
[tex]\[ P_{1996} = P_{1992} \cdot e^{k t_1} \][/tex]
Substituting the known values:
[tex]\[ 153 = 147 \cdot e^{k \cdot 4} \][/tex]
Rearrange to solve for [tex]\( e^{k \cdot 4} \)[/tex]:
[tex]\[ \frac{153}{147} = e^{k \cdot 4} \][/tex]
Simplify the fraction:
[tex]\[ \frac{153}{147} \approx 1.0408 \][/tex]
Now take the natural logarithm of both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ \ln\left(1.0408\right) = k \cdot 4 \][/tex]
[tex]\[ k = \frac{\ln(1.0408)}{4} \][/tex]
[tex]\[ k \approx \frac{0.0395}{4} \][/tex]
Finally, round [tex]\( k \)[/tex] to four decimal places:
[tex]\[ k \approx 0.0100 \][/tex]
4. Estimate the Population in 2008:
Now use the exponential growth formula again to estimate the population in 2008:
[tex]\[ P_{2008} = P_{1992} \cdot e^{k t_{2008}} \][/tex]
Substitute the values:
[tex]\[ P_{2008} = 147 \cdot e^{0.0100 \cdot 16} \][/tex]
Calculate the exponent:
[tex]\[ 0.0100 \cdot 16 = 0.16 \][/tex]
Now calculate [tex]\( e^{0.16} \)[/tex]:
[tex]\[ e^{0.16} \approx 1.1735 \][/tex]
Multiply by the initial population:
[tex]\[ P_{2008} = 147 \cdot 1.1735 \approx 172.5098 \][/tex]
5. Round to the Nearest Million:
Finally, round the estimated population to the nearest million:
[tex]\[ P_{2008} \approx 173 \][/tex]
So, the estimated population of the country in 2008 is approximately 173 million.
1. Understand and Define the Given Values:
- [tex]\( P_{1992} = 147 \)[/tex] million (population in 1992)
- [tex]\( P_{1996} = 153 \)[/tex] million (population in 1996)
- [tex]\( t_1 = 1996 - 1992 = 4 \)[/tex] years (time interval from 1992 to 1996)
- [tex]\( t_{2008} = 2008 - 1992 = 16 \)[/tex] years (time interval from 1992 to 2008)
2. Exponential Growth Formula:
The population growth can be modeled using the formula:
[tex]\[ P = A e^{k t} \][/tex]
where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( A \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the time
3. Calculate the Growth Rate [tex]\( k \)[/tex]:
To find [tex]\( k \)[/tex], we rearrange the formula to solve for [tex]\( k \)[/tex] using the known populations at two different times.
[tex]\[ P_{1996} = P_{1992} \cdot e^{k t_1} \][/tex]
Substituting the known values:
[tex]\[ 153 = 147 \cdot e^{k \cdot 4} \][/tex]
Rearrange to solve for [tex]\( e^{k \cdot 4} \)[/tex]:
[tex]\[ \frac{153}{147} = e^{k \cdot 4} \][/tex]
Simplify the fraction:
[tex]\[ \frac{153}{147} \approx 1.0408 \][/tex]
Now take the natural logarithm of both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ \ln\left(1.0408\right) = k \cdot 4 \][/tex]
[tex]\[ k = \frac{\ln(1.0408)}{4} \][/tex]
[tex]\[ k \approx \frac{0.0395}{4} \][/tex]
Finally, round [tex]\( k \)[/tex] to four decimal places:
[tex]\[ k \approx 0.0100 \][/tex]
4. Estimate the Population in 2008:
Now use the exponential growth formula again to estimate the population in 2008:
[tex]\[ P_{2008} = P_{1992} \cdot e^{k t_{2008}} \][/tex]
Substitute the values:
[tex]\[ P_{2008} = 147 \cdot e^{0.0100 \cdot 16} \][/tex]
Calculate the exponent:
[tex]\[ 0.0100 \cdot 16 = 0.16 \][/tex]
Now calculate [tex]\( e^{0.16} \)[/tex]:
[tex]\[ e^{0.16} \approx 1.1735 \][/tex]
Multiply by the initial population:
[tex]\[ P_{2008} = 147 \cdot 1.1735 \approx 172.5098 \][/tex]
5. Round to the Nearest Million:
Finally, round the estimated population to the nearest million:
[tex]\[ P_{2008} \approx 173 \][/tex]
So, the estimated population of the country in 2008 is approximately 173 million.
