At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
[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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.