Answer:
The amount of years that it takes the population to double is 17.33 years.
Step-by-step explanation:
We have a model for the population of deer, in we know that the population have a net growth rate of 4%.
That is:
[tex]\dfrac{dP}{dt} =0.04P[/tex]
We have to calculate how much time it takes to the population of deers to duplicate its number, given that the growth rate is kept constant.
We can start by solving the first differential equation:
[tex]\dfrac{dP}{dt} =0.04P\\\\\\\int \dfrac{dP}{P}=0.04\int dt\\\\\\ln(P)=0.04t+C_1\\\\\\P(t)=Ce^{0.04t}[/tex]
We don't know the initials conditions to calculate C, but we don't need them to solve this problem.
We will estimate the time h that takes the population to double as:
[tex]P(t+h)=2P(t)\\\\Ce^{0.04(t+h)}=2Ce^{0.04t}\\\\e^{0.04t}\cdot e^{0.04h}=2e^{0.04t}\\\\e^{0.04h}=2\\\\0.04h=ln(2)\\\\h=ln(2)/0.04=17.33[/tex]
The amount of years that it takes the population to double is 17.33 years.
