To determine the tripling time for the alligator population, we need to calculate the time [tex]\( t \)[/tex] at which the initial population will have tripled. In other words, we're looking for the time when [tex]\( P(t) = 3 \cdot P(0) \)[/tex].
Given the population function [tex]\( P(t) = 779 \times 3.25^t \)[/tex]:
1. Let [tex]\( P_0 \)[/tex] be the initial population, so [tex]\( P_0 = 779 \)[/tex].
2. The growth rate is given as [tex]\( 3.25 \)[/tex], meaning the population grows by a factor of 3.25 each year.
3. We are looking for the time [tex]\( t \)[/tex] when [tex]\( P(t) = 3 \cdot P_0 \)[/tex], so:
[tex]\[ 3 \cdot P_0 = P_0 \times 3.25^t \][/tex]
4. Dividing both sides by [tex]\( P_0 \)[/tex] (which is non-zero), we get:
[tex]\[ 3 = 3.25^t \][/tex]
5. To solve for [tex]\( t \)[/tex], we can use logarithms. Taking the natural logarithm of both sides, we get:
[tex]\[ \ln(3) = \ln(3.25^t) \][/tex]
6. Applying the properties of logarithms, we can rewrite it as:
[tex]\[ \ln(3) = t \cdot \ln(3.25) \][/tex]
7. Solving for [tex]\( t \)[/tex], we divide both sides by [tex]\( \ln(3.25) \)[/tex]:
[tex]\[ t = \frac{\ln(3)}{\ln(3.25)} \][/tex]
8. Substitute the values of the natural logarithms:
[tex]\[ \ln(3) \approx 1.0986 \][/tex]
[tex]\[ \ln(3.25) \approx 1.1787 \][/tex]
9. Then:
[tex]\[ t = \frac{1.0986}{1.1787} \approx 0.9321 \][/tex]
Therefore, the tripling time for the alligator population in the swamp is approximately [tex]\( 0.932 \)[/tex] years.
