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Sagot :
To estimate the population of the country in 2014 using the given exponential growth formula, we will follow these steps:
1. Identify the given values:
- Population in 1991 ([tex]\( P_{1991} \)[/tex]): [tex]\( 114 \)[/tex] million
- Population in 1997 ([tex]\( P_{1997} \)[/tex]): [tex]\( 120 \)[/tex] million
- Year 1991 ([tex]\( t_{1991} \)[/tex]): 1991
- Year 1997 ([tex]\( t_{1997} \)[/tex]): 1997
- Year 2014 ([tex]\( t_{2014} \)[/tex]): 2014
2. Calculate the time difference:
- Time difference from 1991 to 1997 ([tex]\( t1 \)[/tex]): [tex]\( t_{1997} - t_{1991} = 1997 - 1991 = 6 \)[/tex] years
- Time difference from 1991 to 2014 ([tex]\( t2 \)[/tex]): [tex]\( t_{2014} - t_{1991} = 2014 - 1991 = 23 \)[/tex] years
3. Set up the exponential growth formula:
Exponential growth formula is given by:
[tex]\[ P = A e^{k t} \][/tex]
Here:
- [tex]\( P \)[/tex] is the future population we are trying to find
- [tex]\( A \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the number of years after the initial year
4. Find the growth rate ([tex]\( k \)[/tex]):
- We have the populations for two different years, so we can use them to find [tex]\( k \)[/tex]:
[tex]\[ P_{1997} = P_{1991} e^{k t1} \][/tex]
Taking natural logs on both sides:
[tex]\[ \ln(P_{1997}) = \ln(P_{1991}) + k t1 \][/tex]
Rearranging to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{\ln(P_{1997}) - \ln(P_{1991})}{t1} \][/tex]
- Substitute the values:
[tex]\[ k = \frac{\ln(120) - \ln(114)}{6} \][/tex]
Calculate the natural logarithms and then divide:
[tex]\[ k \approx \frac{4.7875 - 4.7362}{6} \approx \frac{0.0513}{6} \approx 0.0085489 \][/tex]
- Rounding [tex]\( k \)[/tex] to four decimal places:
[tex]\[ k \approx 0.0085 \][/tex]
5. Calculate the population in 2014:
- Now, we use the exponential growth formula to find the population in 2014:
[tex]\[ P_{2014} = P_{1991} e^{k t2} \][/tex]
- Substitute the values:
[tex]\[ P_{2014} = 114 e^{0.0085 \times 23} \][/tex]
- Calculate the exponent first:
[tex]\[ 0.0085 \times 23 \approx 0.196 \][/tex]
- Exponentiating:
[tex]\[ e^{0.196} \approx 1.216 \][/tex]
- Now, multiply by the initial population:
[tex]\[ P_{2014} \approx 114 \times 1.216 \approx 138.7707 \text{ million} \][/tex]
6. Round the result to the nearest million:
[tex]\[ P_{2014} \approx 139 \text{ million} \][/tex]
Thus, the estimated population in 2014 is 139 million.
[tex]\[\boxed{139}\][/tex]
1. Identify the given values:
- Population in 1991 ([tex]\( P_{1991} \)[/tex]): [tex]\( 114 \)[/tex] million
- Population in 1997 ([tex]\( P_{1997} \)[/tex]): [tex]\( 120 \)[/tex] million
- Year 1991 ([tex]\( t_{1991} \)[/tex]): 1991
- Year 1997 ([tex]\( t_{1997} \)[/tex]): 1997
- Year 2014 ([tex]\( t_{2014} \)[/tex]): 2014
2. Calculate the time difference:
- Time difference from 1991 to 1997 ([tex]\( t1 \)[/tex]): [tex]\( t_{1997} - t_{1991} = 1997 - 1991 = 6 \)[/tex] years
- Time difference from 1991 to 2014 ([tex]\( t2 \)[/tex]): [tex]\( t_{2014} - t_{1991} = 2014 - 1991 = 23 \)[/tex] years
3. Set up the exponential growth formula:
Exponential growth formula is given by:
[tex]\[ P = A e^{k t} \][/tex]
Here:
- [tex]\( P \)[/tex] is the future population we are trying to find
- [tex]\( A \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the number of years after the initial year
4. Find the growth rate ([tex]\( k \)[/tex]):
- We have the populations for two different years, so we can use them to find [tex]\( k \)[/tex]:
[tex]\[ P_{1997} = P_{1991} e^{k t1} \][/tex]
Taking natural logs on both sides:
[tex]\[ \ln(P_{1997}) = \ln(P_{1991}) + k t1 \][/tex]
Rearranging to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{\ln(P_{1997}) - \ln(P_{1991})}{t1} \][/tex]
- Substitute the values:
[tex]\[ k = \frac{\ln(120) - \ln(114)}{6} \][/tex]
Calculate the natural logarithms and then divide:
[tex]\[ k \approx \frac{4.7875 - 4.7362}{6} \approx \frac{0.0513}{6} \approx 0.0085489 \][/tex]
- Rounding [tex]\( k \)[/tex] to four decimal places:
[tex]\[ k \approx 0.0085 \][/tex]
5. Calculate the population in 2014:
- Now, we use the exponential growth formula to find the population in 2014:
[tex]\[ P_{2014} = P_{1991} e^{k t2} \][/tex]
- Substitute the values:
[tex]\[ P_{2014} = 114 e^{0.0085 \times 23} \][/tex]
- Calculate the exponent first:
[tex]\[ 0.0085 \times 23 \approx 0.196 \][/tex]
- Exponentiating:
[tex]\[ e^{0.196} \approx 1.216 \][/tex]
- Now, multiply by the initial population:
[tex]\[ P_{2014} \approx 114 \times 1.216 \approx 138.7707 \text{ million} \][/tex]
6. Round the result to the nearest million:
[tex]\[ P_{2014} \approx 139 \text{ million} \][/tex]
Thus, the estimated population in 2014 is 139 million.
[tex]\[\boxed{139}\][/tex]
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