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^ population of penguins fluctuates from a low of 400 in year / = 0 8,then back to its low of 400 in to a hlgh of 2000 In year year [ra3) (3 points) Assuming the po...

Question

^ population of penguins fluctuates from a low of 400 in year / = 0 8,then back to its low of 400 in to a hlgh of 2000 In year year [ra3) (3 points) Assuming the population of penguins P(t) is a function of time given In the 82ph, find the midline; amplitude; and period of P(t) b} (6 points) Find formula for P(t) .

^ population of penguins fluctuates from a low of 400 in year / = 0 8,then back to its low of 400 in to a hlgh of 2000 In year year [ra 3) (3 points) Assuming the population of penguins P(t) is a function of time given In the 82ph, find the midline; amplitude; and period of P(t) b} (6 points) Find formula for P(t) .



Answers

A population of animals oscillates sinusoidally between a low of 700 on January 1 and a high of 900 on July 1 . (a) Graph the population against time. (b) Find a formula for the population as a function of time, $t,$ in months since the start of the year.

For party. We have toe period. Yes, in the formula two Pi by the baby we know that be equals two, by the way. My no. So the period is 11 or be have to graft. So we started with the pier, which is 11. We know they cost. So it's going star here. Go back to the lowest point into a robot. The initial point to complete one. It's like we'll probably see. We find the monks, which is the average in this case he averages in 100 and 50 plus the amplitude. Jeez, 250. Now we get 1200 for them in your waist truck, my son drown 50 minus 2 50 We get Syrian hungry so that might soon will be 1200 would be 700 Lastly or part of the we have to look a the number 740 which would be room here on the asking, justifying the time for this section. We cannot Broxham into these to me three years if we could use announced autograph later. So part B will approximately be three years

For a party. We need to find the period using that formula toe pie by the baby. We know it would be Sequels to to pine you write a letter and so the period will be for pervy When it the graph on, we'll start with the period. She's 11 years. We know that the graph is causing. So we started at the maximum. We reached the minimum Onda. We go back toe the maximum, thus completing one cycle For part C, we find the maximum which is the average Wally 150 plus the amplitude which is 215 on we get 1200 on for the minimal We so trucked the average minus the amplitude way Get 700 So we have one tallest in 100 That's the maximum 700 in the room 900 as the coverage Lastly for party, we have to find the number of years. Population is less than 100 for so 7 40 Miss, Here we have toe approximate number of years in this section. If we use the natural graphing calculator, we can approximate Thies to be around two years. So we say that for party we will have approximately two years

Oh We have question number five in which there is a craft given. And we have to find the various things about the graph. Okay. So if we look at the graph it is okay. It is like this. This is why This is X. This is -4. This is -1. This is -7. Okay. And 123. This is one, two and three. Okay, so the graph looks like this. Yeah Buffalos like this approximately. Now we have to find first temperatures and midline the period. Now we see that The maximum value is -1 and minimum is -7. So minus one plus minus seven by two. That is -4. So -4 is the midline. So let us start with the midline. A midline Will be equal to what I call to -4. This is the midline then amplitude amplitude is the displacement from the middle line. So be amply cute. Will be the maximum displacement from the middle line. So it will be minus one minus minus four. That is three. Two amplitude equal to three. Let us suppose this to be a part C. The period period. We can see that graph looks like same from here to here and then after it got it starts repeating. So we have a period which will be equal to okay Which should be equal to three. Yes period equals two. Three. Okay party and then find the form for the equation. Okay so we have to find the equation. So it resembles with the sine function like this. So let us suppose the function to be why call to a sign B. X minus C. Publicity. Where is the amplitude? So we have let us support a question number one we have equal to three that is impartial equal to three. Period to buy buy baby Will be equal to three. So we will be equal to two by by three. And see C. Is the feed shift which is equal to zero or horizontal ship ship which is a conclusion because we have started from what exists now the vertical shift D. This will have some value which is distance from this X. Access to the midline. So vertical shift will be equal to minus four but it is plugging in a question number one. It will be why paul too three saying took by by three X minus zero bless minus four. So I called to he's saying two x 2 x three x -4 so they should be the answer. The required a question. Thank you.

So this is the information we're given and I chose to do a sign curve. You can do Signer coastline. So the period equals two pi over B. Let's find the first. So they tell us that the period is four. So I'm going to do to pi over B equals four. Let's cross multiply and solve for B. And we get B equals pi over two. I chose a sign curve. So that's my B value right there. Now the midline is y equals -3, which means it goes down -3. There is that and it tells us the amplitude is too. So we have all of the values in there that we need. So finding the period is the most involved one. Just use the period formula.


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