The population of a species of starfish in the Gulf of Mexico is decreasing at an exponential rate, A(t) = A0e(kt) . Five years ago the population was 10,000. The current population is only 2000. When the population is 500, the starfish population cannot recover. When will this event occur? Show your work and explain the process of solving this problem.

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2015password 2015password · Answer 1 · 2017-09-11T18:01:52+02:00

4 months because it decreases 2000 each year, and 1.5 hundred each month I think so

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facundo3141592 facundo3141592 · Answer 2 · 2019-10-08T03:45:28+02:00

Answer: Hello mate!

we know that the model of the population is described with the function [tex]A(t) = A0e^{kt}[/tex] where A0 and k are constants, and t is time.

we know that five years ago, the population wass 10000 and now is 2000, if we set t = 0 five years ago then, and define t in years:

[tex]A(0) = 10000 = A0e^{k0} = A0[/tex], now we know the value of A0.

and [tex]A(5) = 10000e^{k5} = 2000[/tex]

[tex]e^{k5} = 2000/10000 = 0.2[/tex]

[tex]k5 = ln(0.2)[/tex]

[tex]k = ln(0.2)/5 = -0.32[/tex]

then the equation is:

[tex]A(t) = 10000e^{-0.32t}[/tex]

now we want to find the value of t where A(t) = 500 then:

[tex]A(t) = 500 = 10000e^{-0.32t}[/tex]

[tex]500/10000 = 0.05 = e^{-0.32t}[/tex]

[tex]ln(0.05)/-0.32 = t = 9.4[/tex]

But remember that we took t = 0 five years ago, then the event will ocurr in 9.4 - 5 = 4.4 years.