Answered

Connect with experts and get insightful answers on IDNLearn.com. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.

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]\[ p(t) = 4249 - 1180 \cos \left(\frac{2 \pi}{7} t \right) \][/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 7 years, when will the population size be 4800?

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.)

[tex]\[ t = \][/tex] [tex]$\square$[/tex] years

or

[tex]\[ t = \][/tex] [tex]$\square$[/tex] years

Sagot :

GinnyAnsweridn
To find when the population size will be 4800 during the first 7 years, we need to solve the equation given by:

[tex]\[ p(t) = 4249 - 1180 \cos\left(\frac{2 \pi}{7} t\right) \][/tex]

We set this equal to 4800 to find the time \( t \):

[tex]\[ 4249 - 1180 \cos\left(\frac{2 \pi}{7} t\right) = 4800 \][/tex]

First, solve for \( \cos\left(\frac{2 \pi}{7} t\right) \):

[tex]\[ 4249 - 1180 \cos\left(\frac{2 \pi}{7} t\right) = 4800 \][/tex]

Subtract 4249 from both sides of the equation:

[tex]\[ - 1180 \cos\left(\frac{2 \pi}{7} t\right) = 4800 - 4249 \][/tex]

Simplify the right side:

[tex]\[ - 1180 \cos\left(\frac{2 \pi}{7} t\right) = 551 \][/tex]

Now, divide both sides by -1180 to isolate the cosine term:

[tex]\[ \cos\left(\frac{2 \pi}{7} t\right) = \frac{551}{-1180} \][/tex]

[tex]\[ \cos\left(\frac{2 \pi}{7} t\right) = -\frac{551}{1180} \][/tex]

This further simplifies to:

[tex]\[ \cos\left(\frac{2 \pi}{7} t\right) = -0.4678... \][/tex]

Next, find the inverse cosine of -0.4678 to solve for \( \frac{2 \pi}{7} t \):

[tex]\[ \frac{2 \pi}{7} t = \cos^{-1}(-0.4678) \][/tex]

Using a calculator to find \( \cos^{-1}(-0.4678) \):

[tex]\[ \frac{2 \pi}{7} t \approx 2.2913 \text{ radians} \][/tex]

Now, solve for \( t \):

First instance:
[tex]\[ t = \frac{7}{2 \pi} \cdot 2.2913 \][/tex]

Calculate \( t \):

[tex]\[ t \approx 2.29 \text{ years} \][/tex]

There is another solution because cosine is periodic with period \( 2 \pi \). The other solution within one period (0 to 7 years) would be found as follows:

[tex]\[ \frac{2 \pi}{7} t = 2 \pi - 2.2913 \][/tex]

[tex]\[ \frac{2 \pi}{7} t \approx 3.9919 \text{ radians} \][/tex]

Solve for \( t \):

[tex]\[ t = \frac{7}{2 \pi} \cdot 3.9919 \][/tex]

Calculate \( t \):

[tex]\[ t \approx 4.71 \text{ years} \][/tex]

So, the solutions to the given problem are \( t = 2.29 \) years and \( t = 4.71 \) years.

Thus, the population will be 4800 at:

[tex]\[ t = 2.29 \text{ years} \][/tex]
or
[tex]\[ t = 4.71 \text{ years} \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com for your queries. We’re committed to providing accurate answers, so visit us again soon.