Respuesta :
Solution
- The formula for the reproduction can be derived as follows:
[tex]\begin{gathered} f(1)=50(25)^0 \\ f(2)=50(25)=1250 \\ f(3)=50(25)^2=50(25)^2=31250 \\ \text{ And so on...} \\ \text{ We can generalize for the nth generation as} \\ \\ f(n)=50(25)^{n-1} \\ \\ where, \\ n=\text{ Number of 2month periods} \\ f(n)=\text{ Number of guppies} \end{gathered}[/tex]- With the general formula, we can proceed to solve the question.
- This is done below:
[tex]\begin{gathered} f(n)=10^{10} \\ \\ f(n)=50(25)^{n-1}=10^{10} \\ \frac{100}{2}(25)^{n-1}=10^{10} \\ \\ \text{ Divide both sides by 100. Multiply both sides by 2} \\ 25^{n-1}=2\times(10^{10})\times\frac{1}{100} \\ \\ 25^{n-1}=2\times10^{10}\times10^{-2}=2\times10^8 \\ \\ \frac{25^n}{25}=2\times10^8 \\ \\ 25^n=25(2\times10^8) \\ \text{ Take the natural log of both sides} \\ \\ \ln25^n=\ln(50\times10^8)=\ln(\frac{10^9}{2}) \\ \\ n\ln25=\ln(\frac{10^9}{2}) \\ \\ \text{ divide both sides by }\ln25 \\ \\ n=\frac{\ln(\frac{10^9}{2})}{\ln25} \\ \\ n\approx6.223 \end{gathered}[/tex]- Thus, there are 6.223 2-month periods for the guppies to have 10 billion babies.
- Therefore, the number of months is
[tex]M=2\times6.223=12.44\approx12\text{ months}[/tex]