Answer:
a) [tex]N(t) = 35e^{0.18t}[/tex]
b) The projected population after 6 years is of 103 stray cats.
c) The number of years required for the stray-cat population to reach 700 is 16.64.
Step-by-step explanation:
The population N(t) after t years, following an exponential growth moel, is given by:
[tex]N(t) = N(0)e^{rt}[/tex]
In which N(0) is the initial population and r is the growth rate.
In 1999 the town had 35 stray cats, and the relative growth rate was 18% per year.
This means that [tex]N(0) = 35, r = 0.18[/tex]
(a) Find the function that models the stray-cat population n(t) after t years.
[tex]N(t) = N(0)e^{rt}[/tex]
[tex]N(t) = 35e^{0.18t}[/tex]
(b) Find the projected population after 6 years.
This is N(6).
[tex]N(t) = 35e^{0.18t}[/tex]
[tex]N(6) = 35e^{0.18*6}[/tex]
[tex]N(6) = 103[/tex]
The projected population after 6 years is of 103 stray cats.
(c) Find the number of years required for the stray-cat population to reach 700.
This is t for which N(t) = 700. So
[tex]N(t) = 35e^{0.18t}[/tex]
[tex]700 = 35e^{0.18t}[/tex]
[tex]e^{0.18t} = \frac{700}{35}[/tex]
[tex]e^{0.18t} = 20[/tex]
[tex]\ln{e^{0.18t}} = \ln{20}[/tex]
[tex]0.18t = \ln{20}[/tex]
[tex]t = \frac{\ln{20}}{0.18}[/tex]
[tex]t = 16.64[/tex]
The number of years required for the stray-cat population to reach 700 is 16.64.
