Zoologists are studying two newly discovered species of insects in a previously unexplored section of rain forest. They estimate the current population of insect A to be 1.3 million and the current population of insect B to be 2.1 million. As development is encroaching on the section of rain forest where these insects live, the zoologists estimate the populations of insect A to be reducing at a rate of 3.8% and insect B to be reducing at a rate of 4.6%.

If P represents the population of each species of insect in millions, and t represents the elapsed time in years, then which of the following systems of equations can be used to determine how long it will be before the populations of the two species are equal?

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luisejr77 luisejr77 · Answer 1 · 2019-05-02T05:08:22+02:00

Answer: Obtion B

[tex]P = 1.3e ^ {-0.038t}\\\\P = 2.1e ^ {-0.046t}[/tex]

Step-by-step explanation:

The equation for exponential decay has the following form:

[tex]y = pe ^{-rt}[/tex]

Where

p is the initial population

r is the rate of decrease

t is time.

In this problem we have to:

The current population of insect A to be 1.3 million and the current population of insect B to be 2.1 million.

So

[tex]p_1 = 1.3[/tex] in millions

[tex]p_2 = 2.1[/tex] in millions

We also need the populations of insect to be reduced at a rate of 3.8% and insect to be reduced at a rate of 4.6%.

so:

[tex]r_1 = 0.038\\\\r_2 = 0.046[/tex]

then the exponential decay equation for insect A is:

[tex]P = 1.3e ^ {-0.038t}[/tex]

the exponential decay equation for insect B is:

[tex]P = 2.1e ^ {-0.046t}[/tex]

Finally, the system of equations is:

[tex]P = 1.3e ^ {-0.038t}\\\\P = 2.1e ^ {-0.046t}[/tex]

The answer is the Option B