All right, I have a problem here.
Jacob and Emily ride a ferris wheel at a carnival
in Billings, Montana.
The wheel has a 16-meter diameter.
So let me draw the wheel.
It has a 16-meter diameter.
So let me draw it big.
Give me a lot of space.
So it has a 16-meter diameter, so what's its
radius going to be?
Its radius is going to be half of that, right?
So if I were to draw its radius, just draw it like that.
It's a 16-meter diameter, so it's radius is going to be 8
meters with its lowest point above the ground-- oh,
with its lowest point 1 meter above the ground.
So its lowest point is right here.
This is its lowest point, and that is 1 meter
above the ground.
So this distance right here is 1 meter.
Fair enough.
Assume that Jacob and Emily's height h above the ground is a
sinusoidal function of time where t equals 0 represents the
lowest point on the wheel.
So this is at point t equals 0 right here. t equals 0 is the
lowest point of the wheel.
Write an equation for h.
Oh, I think I forgot, so let me reread it.
Jacob and Emily ride a ferris wheel at a carnival
in Billings, Montana.
The wheel has a 16-meter diameter-- we did
that-- and turns at 3 revolutions per minute.
So it turns at 3 revolutions per minute with its lowest
point 1 meter above the ground.
Assume that Jacob and Emily's height h above the ground is
a sinusoidal function of time, where t equals 0.
So we need to write a function of h, their distance above the
ground, as a function of time, and they're saying that
time is given in seconds.
So, first of all, they're telling us 3 revolutions
every minute, right?
So that's 3 revolutions per 60 seconds, and that's the
same thing as 1 revolution per 20 seconds, right?
I just divide both sides of the per by 3 for 20 seconds.
And one revolution is how many degrees?
One revolution is 360 degrees.
So it's 360 degrees per 20 seconds.
And if you're going 360 degrees per 20 seconds, let's divide--
you know, the per you can just kind of use as an equal
sign of the equation.
360 degrees for 20 seconds.
That means you're going to go what?
18 degrees, Just divide both sides by 20.
18 degrees per second.
And we could have done it with a numerator and a denominator.
3 revs per-- you know, you could have said 3
revs over 60 seconds.
That's actually how I should have done it.
3 revs over 60 seconds is equal to 1 rev over 20 seconds, which
is equal to 360 degrees over 20 seconds, which is equal to 18
degrees per second, right?
So we're going to travel 18 degrees per second.
So the total number of degrees we've traveled in t seconds is
going to be-- so see, if we say the angle, that's the angle
from our starting point.
So let's say we've traveled t seconds, and we're right there.
What is-- let's drop a little altitude right here.
What is this angle going to be, where this angle is right here?
What is this angle going to be?
How many degrees have we traveled?
Well, we say we traveled 18 degrees per second, so if we
travel t seconds, this is going to be 18t degrees, right?
All right, so let's see if we can figure out how their height
as a function of this-- well, as a function of t or as
a function of this angle right here.
So what is this height right here?
Up here?
It's 1 meter plus the radius because this distance
right here is 8.
So we could say that this is-- this point right here is h is
equal to 9 at this point, right?
We could almost view that as the h axis.
So that's h is equal to 9.
So at this point, how high are they?
If this is h-- so right now, let me draw a little
drop and go flat here.
So their height above the ground is this distance h,
which is the same thing as this distance h.
So what is that distance?
Well, it's going to be-- well, if this distance is h, what is
this distance going to be?
This distance is going to be 9 minus h.
How do I know this?
This whole distance is 9.
This distance is h, so-- let me do it in a better color-- so
that this distance right here is 9 minus h.
So let's see what we can do.
What do we know?
We know this distance.
We know this angle is 18t degrees.
And do we know this side?
Sure.
That's the radius.
That's 8.
8 meters.
9 minus h meters, 8 meters, and 18 degrees.
And what are these sides relative to this angle?
Well, if we were to draw a triangle here relative to this
angle right here, the 9 minus h is adjacent, and the 8 meters
is, of course, the hypotenuse, right?
So what trig function deals with adjacent and hypotenuse.
SOHCAHTOA.
CAH, cosine is adjacent over hypotenuse.
So we could say the cosine at 18 degrees, the cosine of 18t
degrees, is equal to its adjacent side.
The adjacent side is 9 minus h.
It's equal to 9 minus h over the hypotenuse, over 8.
And now we can solve for h, and we'll have h
as a function of t.
So we multiply both sides by 8.
You get 8 cosine of 18t is equal to 9 minus h.
Maybe we could subtract 9 from both sides.
So we get minus 9 plus 8 cosine of 18t is equal to minus h, and
then multiply both sides by negative 1, and then you get
9-- positive 9, right-- minus 8 cosine of 18t is equals
to h, or h is equal to 9 minus 8 cosine of 18t.
So there we have it.
We have expressed h as a function of t.
And in the next video, I'm actually going to
graph this function.
See you soon.