Modeling a Cosine Wave with Six Flags Data: Equations and Solutions

[jln1]

Homework Statement

I have a problem, it's last major grade for my pre cal, it's about six flags and here's the information:
H(min)(Trough): 3.ft
H(max)(crest): 32 ft
b= 10 ft
period = 160 ft
write a cosine equation to model of the wave
Assuming still that the leftmost crest intersects the y-axis, write the sine equation to model the wave.

Homework Equations

y= acos(bx)+ d

The Attempt at a Solution

I tried to find amplitude, that's all i can remember to write an cosine or sine equations. The problem that i turned my textbook in and it's due tomorrow. Please help me. Thank you.

 

[QuarkCharmer]

It tells you that the max height will be 32, and the minimum will be 3. If you find the difference between those two, and divide it by 2 you will get your d value, where the graph is shifted up from the X axis. Can you figure out how to get the amplitude from that same information? Just thinking about the definition of amplitude.

 

[jln1]

is the amplitude the highest point of wave from the x-axis? so the a going to be 35?

 

[QuarkCharmer]

If the wave were centered around the x axis, moving equal distances up and down, then you would be right, but in this case, the minimum (trough) of the wave is already at 3. So the whole wave is shifted up some number b.

Take a look at this image from another thread:

The amplitude is "a", and the tan line is the base line about which the whole wave moves, in this example it's shifted up 7.
(Just ignore the rest of the image)

 

[jln1]

So i got d= 14.5, a=17.5?
I guess the equation is 17.5cos(bx)+14.5
Correct me if I'm wrong and what is b and how do i find it? Is it about vertical translation?

 

[QuarkCharmer]

So i got d= 14.5, a=17.5?

You have that reversed a bit.

In your formula:
y= acos(bx)+ d

a is 14.5, meaning the wave moves up and back down 14.5, then down and back up 14.5, all from a height of 17.5. The "d" value, 17.5 is the vertical shift. The "b" value in there changes the frequency. Your question already states that "b" is 10ft.Edit: Was the period (160ft) and the "b" value (10ft) given to you?

The period of a sin/cos wave is 2pi/b, which does not equal 160ft, something is off there.

 

[jln1]

It tells you that the max height will be 32, and the minimum will be 3. If you find the difference between those two, and divide it by 2 you will get your d value, where the graph is shifted up from the X axis. Can you figure out how to get the amplitude from that same information? Just thinking about the definition of amplitude.

As you said, to find d, (32-3)/2=14.5, so the a going to be: 32-14.5= 17.5? Did i misunderstand something here?

 

[QuarkCharmer]

That's my mistake there. In order for the graph to reach a max of 32 and a min of 3, it has to move about a central line that is half way between the two. I don't know what I was thinking when I worded that.

(17.5 + 14.5 = 32) and (17.5 - 14.5 = 3), so 17.5 is the vertical shift (d), and 14.5 is the amplitude.

 

[jln1]

That's OK, thanks for helping me, there's a BOnus: Assuming still that the leftmost crest intersects the y-axis, write the sine equation to model the wave( this is require accounting for a horizontal translation) Would you help me where to start? Is it about the a going to be (-a)?

 

[QuarkCharmer]

You could write it with a negative amplitude sure, there are an infinite number of possibilities for this one. I think the easiest way to go about it would be to consider what it means when the leftmost crest intersects the y axis. That means the wave is in the standard shape of a Cosine Wave! What would you add to the angle of sin(x) to make it look like a cosine wave? (There's an identity for this)

 

[jln1]

pi/2? is it right?

 

[QuarkCharmer]

Ok, so basically you want the graph of sin to start at -pi/2 right? (adding shifts the graph in the negative dir.)

How would you calculate the "phase shift" so it equals -pi/2 ?

 

[jln1]

I actually don't get what you mean
So we have 14.5 cos10x+17.5, are we going to write sine equation that also concern to that cosine equation?

 

[QuarkCharmer]

Of the form
a sin( bx + c) + d

-c/b is equal the the point where the graph starts. If you want your graph to start at -pi/2, you simply set -c/b = -pi/2, and solve for c. All you need to know is the b value. Are you sure that you have the right b value? Like I noted before, your original info says that the period (one full wave) is 160, but the b value is 10...

The period of your wave should be 2pi/b, and 2pi/10 is pi/5, not 160, so there is something afoot there with your original information.

 

[jln1]

yeah well, that must be a problem, I got the information from my friend, I will check again tomorrow. Well, just a few questions and i think i can handle it from here:
+HOw do you find b based on those information above?
+For sine equation? a and d are still the same as at cosine equation? than c is the only we need to find ?
Thank you a lot for your help ^^.

 

[QuarkCharmer]

amplitude and vertical shift will be exactly the same in this situation.

If you want to find b, you need to know the period, or, if you want to find the period, you need to know b. You put the standard period of the trig function over b and that equals the period.

2pi/b = period, depending on the information you have, you solve for b or the period.