Respuesta :
Answer:
27 years.
Step-by-step explanation:
We have been given that the United States population of gray wolves was 1170 in 1991. The population is decreasing by 5% each year.
We can see that population of grey wolves is decreasing exponentially, so we will use an exponential function to model the population after x years.
Since we know that exponential function is in form: [tex]y=a*b^x[/tex], where,
a = Initial value,
b = For decay b is in form (1-r), where r represents decay rate in decimal form.
Let us convert our given decay rate in decimal form.
[tex]5\%=\frac{5}{100}=0.05[/tex]
Upon substituting our given values we will get our function as:
[tex]y=1170(1-0.05)^x[/tex]
[tex]y=1170(0.95)^x[/tex], where x represents number of years.
To find the time it will take the population to reach 300 we will substitute y=300 in our function.
[tex]300=1170(0.95)^x[/tex]
Let us divide both sides of our equation by 1170.
[tex]\frac{300}{1170}=\frac{1170(0.95)^x}{1170}[/tex]
[tex]0.2564102564102564=(0.95)^x[/tex]
Let us take natural log of both sides of our equation.
[tex]ln(0.2564102564102564)=ln((0.95)^x)[/tex]
Using natural log property [tex]ln(a^b)=b*ln(a)[/tex] we will get,
[tex]ln(0.2564102564102564)=x*ln(0.95)[/tex]
[tex]\frac{ln(0.2564102564102564)}{ln(0.95)}=\frac{x*ln(0.95)}{ln(0.95)}[/tex]
[tex]\frac{-1.3609765531356007834}{-0.0512932943875505}=x[/tex]
[tex]x=26.53322\approx 27[/tex]
Therefore, it will take approximately 27 years to the population of grey wolves to reach 300.
