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:
, 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.
![5\%=\frac{5}{100}=0.05](https://tex.z-dn.net/?f=5%5C%25%3D%5Cfrac%7B5%7D%7B100%7D%3D0.05)
Upon substituting our given values we will get our function as:
![y=1170(1-0.05)^x](https://tex.z-dn.net/?f=y%3D1170%281-0.05%29%5Ex)
, 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.
![300=1170(0.95)^x](https://tex.z-dn.net/?f=300%3D1170%280.95%29%5Ex)
Let us divide both sides of our equation by 1170.
![\frac{300}{1170}=\frac{1170(0.95)^x}{1170}](https://tex.z-dn.net/?f=%5Cfrac%7B300%7D%7B1170%7D%3D%5Cfrac%7B1170%280.95%29%5Ex%7D%7B1170%7D)
![0.2564102564102564=(0.95)^x](https://tex.z-dn.net/?f=0.2564102564102564%3D%280.95%29%5Ex)
Let us take natural log of both sides of our equation.
![ln(0.2564102564102564)=ln((0.95)^x)](https://tex.z-dn.net/?f=ln%280.2564102564102564%29%3Dln%28%280.95%29%5Ex%29)
Using natural log property
we will get,
![ln(0.2564102564102564)=x*ln(0.95)](https://tex.z-dn.net/?f=ln%280.2564102564102564%29%3Dx%2Aln%280.95%29)
![\frac{ln(0.2564102564102564)}{ln(0.95)}=\frac{x*ln(0.95)}{ln(0.95)}](https://tex.z-dn.net/?f=%5Cfrac%7Bln%280.2564102564102564%29%7D%7Bln%280.95%29%7D%3D%5Cfrac%7Bx%2Aln%280.95%29%7D%7Bln%280.95%29%7D)
![\frac{-1.3609765531356007834}{-0.0512932943875505}=x](https://tex.z-dn.net/?f=%5Cfrac%7B-1.3609765531356007834%7D%7B-0.0512932943875505%7D%3Dx)
![x=26.53322\approx 27](https://tex.z-dn.net/?f=x%3D26.53322%5Capprox%2027)
Therefore, it will take approximately 27 years to the population of grey wolves to reach 300.