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>> I will just start solving this problem
by sketching a little graph here. That's the ground level
and the diameter of this London Eye, the Ferris wheel,
that we are looking at, is 135 meters.
That means the radius is half of 135 meters, that's very big.
It takes half an hour for it to spin one revolution
and when t equals 0 minutes, that's the point on the ground
where people get on the Ferris wheel;
and when t equals 15 minutes,
that's the highest point of London Eye.
We write down the general form
of the sinusoid, s of t equals A sine
of (omega t plus phi) plus B. And we know that a is 135 over 2.
We simply get A by finding the radius of the Ferris wheel,
and B also equals 135 over 2; we find B by finding the distance
between the ground level and the center of the Ferris wheel.
In this case, since the people can get on the Ferris wheel
on the ground level, then B is the same as the radius.
We then substitute A and B into general form
of the sinusoid, we get (135 over 2) times sine
of (omega t plus phi) plus (135 over 2).
We substitute when t equals 0 in there,
that's at the ground level,
then the y value should be 0 at this point.
So we have (135 over 2) times sine of (phi) plus (135 over 2) equals 0.
We subtract 135 over 2 on both sides and divide both sides
by 135 over 2 to get sine of phi equals negative 1
and phi equals negative pi over 2.
We substitute it back into the equation we have;
we get s of t equals (135 over 2) times sine of (omega t minus pi
over 2) plus (135 over 2).
Now we consider the highest point.
We still have to solve for omega.
When t equals 15, it will be at the highest point
of the Ferris wheel, which is 135 meters from the ground.
We have (135 over 2) sine
of (15 omega minus pi-halves) plus (135 over 2) equals 135.
We multiply both sides of the equation by 2 to get 135 sine
of (15 omega minus pi- halves) plus 135 equals 270.
Now we know that 135 plus 135 is 270, then we have sine
of (15 omega minus pi-halves) equals 1.
15 omega minus pi-halves must equal to pi-halves.
15 omega equals phi, and omega equals pi over 15.
Then, substitute it back in to get our answer.
s of t equals (135 over 2) sine of (pi times t
over 15 minus pi-halves) plus 135 over 2.
over 15 minus pi-halves) plus 135 over 2.