Naturalists find that the population of some kinds of predat

[Jenkins]

Here's the question:

Naturalists find that the population of some kinds of predatory animals vary periodically, Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when time t= 0 years. A minimum number, 200 foxes occured when t = 2.9 years. The next maximum, 800 foxes, occured at t = 5.1 year.

I need to draw this sin graph and get the equation expressing the number of foxes as a function of time t.

I have a slight idea of how this graph should look like.. It would be a sin graph starting at (0, 0), then moving from (2.9, 200), then (5.1, 800), then (7.3, 200), then (10.2, 0), then (13.1, -200), and then (15.3, -800). Is this right? :?

I have no clue about the equation though. I wrote down the following:

. . .y = 300 + sin(1.43x + 0.4) + 500

I think this is the correct equation.. But I don't understand it. Is it possible that someone could help?

Thanks in advance.

 

[stapel]

I think this is the correct equation.. But I don't understand it.

How did you create this equation? (I don't see how you wouldn't understand something you yourself thought up and created, is why I ask.)

Thank you.

Eliz.

 

[Jenkins]

Oh no, I came at the end of class and my teacher told me to copy down the equation and that it would help. Im not sure if I copied it correctly though because I was rushed :? ..

I don't really understand how to find the period or the amplitude of this graph.. Would the amplitude be 800?

 

[stapel]

The format, "300 + (sine of something) + 500", seems a tad improbable.

I'm not sure you arrived at your various graph points, especially the ones that require populations to be negative...? Please reply with your reasoning.

You are given that the minimum population is 200 and the maximum is 800. What then is the amplitude and "centerline"?

You are given that the time difference between the max and min points was 5.1 - 2.9 years. What then is the period of the function?

Thank you.

Eliz.

 

[Jenkins]

To find amplitude.. would it be 1/2 (800-200) = 300?

I don't understand how to find the period though.. Wouldn't you divide something by 2.2?

Edit: Thanks for the reply again

 

[stapel]

Since you start at 200 foxes and end at 800 foxes.. Would your amplitude be 400?

Is 400 half the difference of 800 and 200?

And the center line is.. 600?

Is 600 midway between 200 and 800?

I don't understand how to find the period though.

The "period" is how long the cycle takes to repeat. If min-to-max is 2.2 years, then how long is an entire cycle (min-to-max, and back to min)?

Have trig functions not been covered in class yet...?

Eliz.

 

[Jenkins]

Sorry, I just edited my post right when you posted your new one.
You would find the amplitude by 1/2 (800-200) = 1/2 (600) = 300

Would the period be 8.8? Since it's a sin graph.. It goes 2.2+2.2+2.2+2.2? Which would be 8.8.. Right?

 

[stapel]

Would the period be 8.8?

The "period" is how long the cycle takes to repeat. If min-to-max is 2.2 years, then how long is an entire cycle (min-to-max, and then back to min)?

Eliz.

 

[Jenkins]

From min to max would be 2.2. Then back to min would be 4.4? Then you take 2pi/4.4? Which equals to 1.42799 which rounds to 1.43? Would that be why it's 1.43?

And.. Since you can't have a negative population of foxes.. It vertical shift would be at 500 because the amplitude is at 300. So once you go down 300 from 500, you would get to 200.. Then when go up from 500 by 300, it gets to 800? Wouldn't that be a negative sin graph though because the graph goes down and then up?

I still don't understand the horizontal shift of .4 though..

 

[stapel]

From min to max would be 2.2. Then back to min would be 4.4?

Yes; the entire cycle is 4.4 years.

Then you take 2pi/4.4?

Why? What does that do for you?

[The] vertical shift would be at 500...

Yes. So the function will be something along the lines of y(t) = 300sin(something or other) + 500.

Wouldn't that be a negative sin graph though...

Or one with a sufficient phase shift. Either way is fine.

I still don't understand the horizontal shift of 0.4 though..

How did you derive this value?

Thank you.

Eliz.

 

[Jenkins]

Oh.. In my book it says..

Let b be a positive real number. The period of y=a sin(bx) is 2pi/b

Would b be 4.4? Or does the above formula not apply here

Oh the .4 horizontal shift was from the formula I copied down while rushing.. but of course that formula hasn't been so reliable so nevermind.

Edit: I was wondering though.. Is there a horizontal shift to this problem? If so, How would I find it.. I flipped through my book but it's explanation is really vague and doesn't apply to word problems.