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Sagot :
To estimate the population of a country in the year 2016, using its population data from the years 1994 and 1998, we can apply the exponential growth formula:
[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]\( k \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years.
### Step 1: Identify the known values
- Initial population in 1994: [tex]\( A = 76 \)[/tex] million
- Population in 1998: [tex]\( P = 81 \)[/tex] million
- The time span between 1994 and 1998: [tex]\( t = 1998 - 1994 = 4 \)[/tex] years
- Target year for population estimate: 2016
### Step 2: Find the growth rate [tex]\( k \)[/tex]
First, we rearrange the exponential growth formula to solve for the growth rate [tex]\( k \)[/tex]:
[tex]\[ P = A e^{kt} \][/tex]
[tex]\[ \frac{P}{A} = e^{kt} \][/tex]
[tex]\[ \ln\left(\frac{P}{A}\right) = kt \][/tex]
[tex]\[ k = \frac{1}{t} \ln\left(\frac{P}{A}\right) \][/tex]
Substituting the known values:
[tex]\[ P = 81 \, \text{million} \][/tex]
[tex]\[ A = 76 \, \text{million} \][/tex]
[tex]\[ t = 4 \, \text{years} \][/tex]
[tex]\[ k = \frac{1}{4} \ln\left(\frac{81}{76}\right) \][/tex]
From the result, we know:
[tex]\[ k \approx 0.015928953596526945 \, \text{(per year)} \][/tex]
### Step 3: Estimate the population in 2016
Now we use the exponential growth formula again to find the population in 2016.
Let's calculate the time span from 1994 to 2016:
[tex]\[ t_{\text{target}} = 2016 - 1994 = 22 \, \text{years} \][/tex]
Using the exponential growth formula:
[tex]\[ P = A e^{kt_{\text{target}}} \][/tex]
Substitute the values:
[tex]\[ A = 76 \, \text{million} \][/tex]
[tex]\[ k \approx 0.015928953596526945 \, \text{(per year)} \][/tex]
[tex]\[ t_{\text{target}} = 22 \, \text{years} \][/tex]
[tex]\[ P = 76 \cdot e^{0.015928953596526945 \cdot 22} \][/tex]
From the result, we know:
[tex]\[ P \approx 107.89627181144573 \, \text{million} \][/tex]
Rounding to the nearest million:
[tex]\[ P = 108 \, \text{million} \][/tex]
### Answer:
The estimated population in 2016 is [tex]\(\boxed{108 \, \text{million}}\)[/tex].
[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]\( k \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years.
### Step 1: Identify the known values
- Initial population in 1994: [tex]\( A = 76 \)[/tex] million
- Population in 1998: [tex]\( P = 81 \)[/tex] million
- The time span between 1994 and 1998: [tex]\( t = 1998 - 1994 = 4 \)[/tex] years
- Target year for population estimate: 2016
### Step 2: Find the growth rate [tex]\( k \)[/tex]
First, we rearrange the exponential growth formula to solve for the growth rate [tex]\( k \)[/tex]:
[tex]\[ P = A e^{kt} \][/tex]
[tex]\[ \frac{P}{A} = e^{kt} \][/tex]
[tex]\[ \ln\left(\frac{P}{A}\right) = kt \][/tex]
[tex]\[ k = \frac{1}{t} \ln\left(\frac{P}{A}\right) \][/tex]
Substituting the known values:
[tex]\[ P = 81 \, \text{million} \][/tex]
[tex]\[ A = 76 \, \text{million} \][/tex]
[tex]\[ t = 4 \, \text{years} \][/tex]
[tex]\[ k = \frac{1}{4} \ln\left(\frac{81}{76}\right) \][/tex]
From the result, we know:
[tex]\[ k \approx 0.015928953596526945 \, \text{(per year)} \][/tex]
### Step 3: Estimate the population in 2016
Now we use the exponential growth formula again to find the population in 2016.
Let's calculate the time span from 1994 to 2016:
[tex]\[ t_{\text{target}} = 2016 - 1994 = 22 \, \text{years} \][/tex]
Using the exponential growth formula:
[tex]\[ P = A e^{kt_{\text{target}}} \][/tex]
Substitute the values:
[tex]\[ A = 76 \, \text{million} \][/tex]
[tex]\[ k \approx 0.015928953596526945 \, \text{(per year)} \][/tex]
[tex]\[ t_{\text{target}} = 22 \, \text{years} \][/tex]
[tex]\[ P = 76 \cdot e^{0.015928953596526945 \cdot 22} \][/tex]
From the result, we know:
[tex]\[ P \approx 107.89627181144573 \, \text{million} \][/tex]
Rounding to the nearest million:
[tex]\[ P = 108 \, \text{million} \][/tex]
### Answer:
The estimated population in 2016 is [tex]\(\boxed{108 \, \text{million}}\)[/tex].
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