Suppose that poaching reduces the population of an endangered animal by
8% per year. Further suppose that when the population of this animal falls below
70, its extinction is inevitable (owing to the lack of reproductive options without severe in-breeding). If the current population of the animal is
1700, when will it face extinction? Comment on the validity of this exponential model.

We have been given that the population of an endangered animal by reduces 8% per year. the current population of the animal is 1700. When the population of this animal falls below 70, its extinction is inevitable.

Let us write the model for population of this animal (y) after x years.

Since we know that an exponential decay function is in form: [tex]y=a*(1-r)^x[/tex], where,

a= Initial value.

r = Rate in decimal form.

Let us convert our given rate in decimal form.

[tex]8\%=\frac{8}{100}=0.08[/tex]

Upon substituting a= 1700 and r = 0.08 in exponential function, we will get the model of animal population as:

[tex]y=1700*(1-0.08)^x[/tex]

[tex]y=1700*(0.92)^x[/tex]

Let us find, when the animal population will face extinction by substituting y=70 in our function.

[tex]70=1700*(0.92)^x[/tex]

[tex]\frac{70}{1700}=\frac{1700*(0.92)^x}{1700}[/tex]

[tex]0.0411764705882353=0.92^x[/tex]

Let us take natural log of both sides of our equation.

[tex]ln(0.0411764705882353)=ln(0.92^x)[/tex]

[tex]ln(0.0411764705882353)=x*ln(0.92)[/tex]

[tex]-3.1898882879949483163=x*-0.0833816089390511[/tex]

[tex]x=\frac{-3.1898882879949483163}{-0.0833816089390511}[/tex]

[tex]x=38.2564971890460849445284\approx 38.26[/tex]

Therefore, the population of the animal will face extinction after 38.26 years.