To estimate the population in 2005 using the exponential growth formula, follow these steps:
Given data:
- Population in 1991, [tex]\( P_{1991} = 15 \)[/tex] million
- Population in 2001, [tex]\( P_{2001} = 16 \)[/tex] million
- Time difference between 1991 and 2001, [tex]\( t = 2001 - 1991 = 10 \)[/tex] years
The exponential growth formula is:
[tex]\[ P(t) = P_0 e^{k t} \][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
Step 1: Solve for [tex]\( k \)[/tex]
The growth formula between 1991 and 2001 is:
[tex]\[ P_{2001} = P_{1991} e^{k t} \][/tex]
Substitute the given values:
[tex]\[ 16 = 15 e^{k \cdot 10} \][/tex]
Divide both sides by 15:
[tex]\[ \frac{16}{15} = e^{10k} \][/tex]
Take the natural logarithm on both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ \ln\left(\frac{16}{15}\right) = 10k \][/tex]
[tex]\[ k = \frac{\ln\left(\frac{16}{15}\right)}{10} \][/tex]
Calculate [tex]\( k \)[/tex] and round to four decimal places:
[tex]\[ k \approx 0.0065 \][/tex]
Step 2: Estimate the population in 2005
The population formula is now:
[tex]\[ P_{2005} = P_{1991} e^{k \cdot 14} \][/tex]
where the time difference from 1991 to 2005 is [tex]\( t = 2005 - 1991 = 14 \)[/tex] years.
Substitute the known values:
[tex]\[ P_{2005} = 15 e^{0.0065 \cdot 14} \][/tex]
Calculate the exponent:
[tex]\[ 0.0065 \cdot 14 = 0.091 \][/tex]
So, now we have:
[tex]\[ P_{2005} = 15 e^{0.091} \][/tex]
Calculate the value of [tex]\( e^{0.091} \)[/tex]:
[tex]\[ e^{0.091} \approx 1.0954 \][/tex]
Now substitute this value back:
[tex]\[ P_{2005} = 15 \cdot 1.0954 \approx 16.418 \][/tex]
Finally, round the estimated population to the nearest million:
[tex]\[ P_{2005} \approx 16 \][/tex]
So, the estimated population in 2005 is 16 million.
