CMIIh

2021-08-16

A report tells us that in 2009, there were 870 gray wolves in Idaho, but that the population declined by 19% annual rate of decrease continues.
a) Find an exponential model that gives the wolf population W as a function of the time t in years since 2009.
$W=870{\left(1-.19\right)}^{t}$
b) It is expected that the wolf population cannot recover if there are fewer than 25 individuals. How long must this rate of decline continue for the wolf population to reach 25?

Nathanael Webber

Given: The number of gray wolves in Idalho in year 2009 was 870.
It is given that population decreases annually at rate 19%.
To find:
a) To find an exponential models that gives wolves population W as a function of time t in years since 2009.
b) Number of years in which population of wolves reaches to 25.
Solution: We know that when population decline continuosly then population after time t years is given as:
$W={P}_{0}{\left(1-r\right)}^{t}$, where W is population after time t years, P_0 is initial population and r is rate of decline
We have given, ${P}_{0}=870$
$r=19\mathrm{%}$
$=\frac{19}{100}$
$=0.19$
Therefore, an exponential models that gives wolves population is:
$W=870{\left(1-0.19\right)}^{t}$
$=870{\left(0.81\right)}^{t}$
Now, we have to find t when W=25
$25=870{\left(0.81\right)}^{t}$
${\left(0.81\right)}^{t}=\frac{25}{870}$
${\left(0.81\right)}^{t}=0.028$
${\mathrm{log}\left(0.81\right)}^{t}=\mathrm{log}\left(0.028\right)$
$t\cdot \mathrm{log}\left(0.81\right)=\mathrm{log}\left(0.028\right)$
$t\cdot \left(-0.091\right)=-1.55$
$t=\frac{-1.55}{-0.091}$
$t=17.03$
Therefore, number of years the population will declined to 25 is 17.03 years.

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