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The population sizes of many animal species rise and fall over time. Suppose that the population size of a certain species can be modeled by the following.

[tex]p(t) = 4096 - 1269cos(\frac{2\pi }{11} t)[/tex]

In this equation, p(t) represents the total population size and t is the time (in years). Suppose we start at t = 0 years.

During the first 11 years, when will the population size be 4700?

Do not round any intermediate computations, and round your answer(s) to the nearest hundredth of a year. (If there is more than one answer, enter additional answers with the "or" button.)

There are two answers

Respuesta :

Solving the trigonometric equation, it is found that during the first 11 years, the population size will be of 4700 in 3.62 and 7.24 years.

What is the trigonometric equation for the population?

It is given by:

[tex]p(t) = 4096 - 1269\cos{\left(\frac{2\pi}{11}t\right)}[/tex]

It will be of 4700 when p(t) = 4700, hence:

[tex]p(t) = 4096 - 1269\cos{\left(\frac{2\pi}{11}t\right)}[/tex]

[tex]4700 = 4096 - 1269\cos{\left(\frac{2\pi}{11}t\right)}[/tex]

[tex]604 = - 1269\cos{\left(\frac{2\pi}{11}t\right)}[/tex]

[tex]\cos{\left(\frac{2\pi}{11}t\right)} = -\frac{604}{1269}[/tex]

[tex]\cos{\left(\frac{2\pi}{11}t\right)} = -0.47596532702[/tex]

[tex]\cos^{-1}{\cos{\left(\frac{2\pi}{11}t\right)}} = cos^{-1}{-0.47596532702}[/tex]

Then:

[tex]\frac{2\pi}{11}t = 2.06685766[/tex]

[tex]t = \frac{11 \times 2.06685766}{2\pi}[/tex]

[tex]t = 3.62k, k = 1, 2, ...[/tex]

Thus:

t = 3.62 x 1 = 3.62

t = 3.62 x 2 = 7.24.

The population size will be of 4700 in 3.62 and 7.24 years.

More can be learned about trigonometric equations at https://brainly.com/question/24680641

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