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Answer :
To solve this problem, we're working with an exponential growth (or decay) model. The formula we use for exponential growth or decay is:
[tex]\[ P(t) = P_0 \cdot e^{kt} \][/tex]
Where:
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex].
- [tex]\( k \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years.
- [tex]\( e \)[/tex] is Euler's number, approximately equal to 2.71828.
In this problem, we are given:
- The initial population in 2005, [tex]\( P_0 = 36.3 \)[/tex] million.
- The projected population in 2029, [tex]\( P(t) = 18.7 \)[/tex] million.
- The number of years between 2005 and 2029, [tex]\( t = 2029 - 2005 = 24 \)[/tex] years.
Our task is to find the growth rate, [tex]\( k \)[/tex].
First, we rearrange the exponential growth formula to solve for [tex]\( k \)[/tex]:
[tex]\[ P(t) = P_0 \cdot e^{kt} \][/tex]
[tex]\[ \frac{P(t)}{P_0} = e^{kt} \][/tex]
[tex]\[ \ln\left(\frac{P(t)}{P_0}\right) = kt \][/tex]
[tex]\[ k = \frac{1}{t} \cdot \ln\left(\frac{P(t)}{P_0}\right) \][/tex]
Now, substitute the given values into the formula:
1. Calculate the ratio of the future population to the initial population:
[tex]\[ \frac{18.7}{36.3} \][/tex]
2. Take the natural logarithm:
[tex]\[ \ln\left(\frac{18.7}{36.3}\right) \][/tex]
3. Divide the result by the number of years (24):
[tex]\[ k = \frac{1}{24} \cdot \ln\left(\frac{18.7}{36.3}\right) \][/tex]
After performing these calculations, you will find that the projected growth rate, [tex]\( k \)[/tex], is approximately:
[tex]\[ k \approx -0.0276 \][/tex]
(Rounded to four decimal places.) This indicates a negative growth rate, which means the population is projected to decrease over the given time period.
[tex]\[ P(t) = P_0 \cdot e^{kt} \][/tex]
Where:
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex].
- [tex]\( k \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years.
- [tex]\( e \)[/tex] is Euler's number, approximately equal to 2.71828.
In this problem, we are given:
- The initial population in 2005, [tex]\( P_0 = 36.3 \)[/tex] million.
- The projected population in 2029, [tex]\( P(t) = 18.7 \)[/tex] million.
- The number of years between 2005 and 2029, [tex]\( t = 2029 - 2005 = 24 \)[/tex] years.
Our task is to find the growth rate, [tex]\( k \)[/tex].
First, we rearrange the exponential growth formula to solve for [tex]\( k \)[/tex]:
[tex]\[ P(t) = P_0 \cdot e^{kt} \][/tex]
[tex]\[ \frac{P(t)}{P_0} = e^{kt} \][/tex]
[tex]\[ \ln\left(\frac{P(t)}{P_0}\right) = kt \][/tex]
[tex]\[ k = \frac{1}{t} \cdot \ln\left(\frac{P(t)}{P_0}\right) \][/tex]
Now, substitute the given values into the formula:
1. Calculate the ratio of the future population to the initial population:
[tex]\[ \frac{18.7}{36.3} \][/tex]
2. Take the natural logarithm:
[tex]\[ \ln\left(\frac{18.7}{36.3}\right) \][/tex]
3. Divide the result by the number of years (24):
[tex]\[ k = \frac{1}{24} \cdot \ln\left(\frac{18.7}{36.3}\right) \][/tex]
After performing these calculations, you will find that the projected growth rate, [tex]\( k \)[/tex], is approximately:
[tex]\[ k \approx -0.0276 \][/tex]
(Rounded to four decimal places.) This indicates a negative growth rate, which means the population is projected to decrease over the given time period.
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