In the last part of the problem we figured out that the
function of the height of the ferris wheel, the people at the
ferris wheel at any time is a function of t. h of t is equal
to 9 minus 8 cosine of 18t where t is in seconds.
Now the second part of this problem they want us to graph h
as a function of t between 0 is less than or equal to t, is
less than or equal to 30.
So let me draw axes.
So let's say that that's my h-axis.
Let's say that this is my t-axis.
So this is t equals 0, and this is t is equal to 30 seconds.
I get confused when I see this 18 here or whatever.
So what I'm going to do first of all is I'm going to graph a
different function, slightly different function, then I'll
translate it to this function.
I'm going to graph h of theta is equal to 9
minus 8 cosine of theta.
I think you'll see where I'm going with this
when I'm all done.
So let's try to graph h of theta is equal to 9
minus 8 cosine of theta.
So when t is equal to 30 seconds, what
is theta equal to?
So 30 times 18, that's 540.
So this is 540 degrees.
Same thing.
I'll write the thetas in red above the t-axis.
This is 540 degrees, so that's like two times
around the circle.
So that's 540 degrees, then this is going to
be roughly 270 degrees.
So, 90 degrees will be about 1/3 of this.
That would be 90 degrees, that would be 180, so that would be
90 degrees, that would be 180 degrees, this would be 360
degrees, and this would be 360 plus 90 so this will
be 450 degrees.
If you wanted to figure out the corresponding time,
you just take this degree and divide by 18.
So it's 90 divided by 18 is what?
It's five, right?
So if I were to write here, this is at 5 seconds, this is
at 10 seconds, this is 15 seconds, this is 20 seconds,
this is -- sorry, this is 25 seconds, this is 30 seconds.
Actually, a simple thing we could do is let's just figure
out what the value of the function is at these points.
Because these are pretty easy degrees to figure out
what the cosine value is.
So let's figure out -- let me draw a table.
Tables are always good and I'll do it in yellow.
So I'll draw a t theta and h.
This might be kind of an unconventional way of doing
things, but I have a simple mind so this is actually
how I like to do it.
So I like to think of theta as 0, 90, 180,
270, 360, 450 and 540.
And t, the corresponding time of those, as 0,
5, 10, 15, 20, 25, 30.
It's not rocket science here.
When t equals 15 seconds, 15 times 18, we're trying
to find the cosine of 270 degrees, right?
15 times 18 is 270 degrees.
I'm just doing this because I don't have a calculator and
this will help me pick good points.
So when t is equal to 0, what is height?
Or t is equal to 0, theta is equal to 0, so cosine of theta
is -- cosine of 0 is 1.
So 9 minus 8 is 1.
I'm going to do h in a different color.
So this is 1.
Cosine of 90 degrees?
Cosine of 90 degrees is 0.
So 9 minus 0 is 9.
Cosine of 180 degrees?
So we're going all the way around the unit circle.
Cosine of 180 degrees is minus 1.
So minus 1 times minus 8 is plus 8, so 9 plus 8, that's 15.
Cosine of 270 degrees are pointing straight down, so the
x-coord is going to be 0.
So once again, we're at 9 again.
9 minus 0.
360 degrees.
Cosine of 360 degrees is the same thing as
cosine of 0, right?
So once again, I mean we've gone around the circle once.
So it's going to be the same as 0, so it's going to be 1.
And 450 is going to be the same thing as 90.
So it's going to be 9 and then 15 degrees.
So let's plot these points.
Actually, let me just draw 15 up here.
So what are the points that keep showing up?
So this is 1, that's 1, that's 1, and then we have 9.
1, there's 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.
Fair enough.
So let me draw some guidelines just to help us.
Actually, let me do them kind of hard to see, because I
don't want to draw too much attention to the guidelines.
I could do one guideline there.
Then I'll do a bottom guideline right there.
Then the 9 keeps showing up.
Oh, you know what, I can't add.
What's 9 plus 8?
It's not 15, it's 17.
Sorry, clearly, I need to practice my addition.
So this is 9 plus 8, this is 17.
And I realized that because I was like, well 9 should be in
the middle, so this is actually 17.
Ignore my little marks here.
That's 17, this is 1.
Ignore the marks.
9 would be right in the middle between 1 and 17.
So let me draw kind of mediant point right there.
So this is 9.
Sorry I can't add properly.
Then let's draw the graph or at least plot the
points on the graph.
So, t equals 0 where h equals 1, so that's this point.
That's right here.
When t equals 5, h is equal to 9, right here.
When t is equal to 10, h is equal to 17.
When t is equal to 15, h is 9 again.
So it's right here.
At 20 we're back at 1.
I think you see the pattern.
At 25 we're back at 9.
And then at 30 we're back at 17, not 15, because now I
have corrected my mistake.
And this is going to be sined graph, it's going to look
something like this.
Let me do it in a vibrant color so I can overwrite everything
and it's going to look something like this.
Go oops, and then up and them down here.
Curve up, come back down, curve up and then come back down.
Like that.
So that's our graph.
I think in the problem they tell us to approximate.
Actually, let me open up my cousin's problem -- my other
account has timed-out on me while I recorded this.
They wanted to approximate when t equals 4 what the height is.
So when t equals 4 the height is like right
around there, right?
So the height is a little bit less than 9.
And I don't know, 7 or 8 meters in the air.
And when time is equal to 10 -- well, time equal 10, we figured
out exactly, we know that there are 17 meters in the air.
So I know this was kind of a little messy and graphing
trick functions tend to be, but hopefully you found
this vaguely useful.
Have fun.