The answer is:
It will take 20.5 months to the populations to be equal.
Since from the statement we know that both populations are growing, we need to use the formula to calculate the exponential growth.
The exponential growth is defined by the following equation:
[tex]P(t)=StartPopulation*e^{\frac{growthpercent}{100}*t}[/tex]
Now,
Calculating for the rabbits, we have:
[tex]StartPopulation=20\\GrowthPercent=5\\[/tex]
So, writing the equation for the rabbits, we have:
[tex]P(t)=20*e^{\frac{5}{100}*t}[/tex]
[tex]P(t)=20*e{0.05}*t}[/tex]
Calculating for the fox, we have:
[tex]StartPopulation=30\\GrowthPercent=3\\[/tex]
So, writing the equation for the fox, we have:
[tex]P(t)=30*e{\frac{3}{100}*t}[/tex]
[tex]P(t)=30*e^{0.03}*t}[/tex]
Then, if we want to calculate how long does it takes for these populations to be equal, we need to make their equations equal, so:
[tex]20*e^{0.05}*{t}=30*e^{0.03}*{t}\\\\\frac{20}{30}=\frac{e^{0.03}*{t}}{e^{0.05}*t}}\\\\0.66=e^{0.03t-0.05t}=e^{-0.02t}\\\\0.66=e^{-0.02t}\\\\ln(0.66)=ln(e^{-0.02t})\\\\-0.41=-0.02t\\\\t=\frac{-0.41}{-0.02}=20.5[/tex]
Hence, we have that it will take 20.5 months to the populations to be equal.
