SOLUTION: Suppose that the height h(t) (in meters) of a cabin of the London Eye as a function of the time t (in minutes) can be written as h(t)=a + b * cos(c * t) wit

Question 1069701: Suppose that the height h(t) (in meters) of a cabin of the London Eye as a function of the time t (in minutes) can be written as
h(t)=a + b * cos(c * t)
with a, b and c constants.
At time t=0 the cabin should start at the bottom. Use the data above to find the coefficients a, b and c. For c, work in radians. Give exact answers.
Below is the answer provided with my questions <<< INTERSPERSED >>>:
<<< I CAN'T VISUALIZE HOW COS COMES INTO THIS -- BY CONTRAST I THINK I COULD SEE USING SIN CAUSE = OPP/HYP, SO MULTIPLYING IT BY RADIUS (WHICH IS HYP) WOULD GIVE OPP, WHICH WOULD BE THE Y-COMPONENT OR THE DIFFERENCE OR ADDITION IN HEIGHT FROM OR TO THE HEIGHT OF THE AXIS>>>
It takes 30 minutes for a complete tour. This implies that 30* c = 2 * pi;
<<< IN 30 MIN. THEY TRAVEL ONE REVOLUTION OR 2 * PI. OK. BUT WHY DOES THIS MEAN THAT THE VALUE IN THE COS PARENTHESES HAS TO EQUAL THAT 2 * PI?
AND WHAT HAPPENED TO A & B ... WHAT IS THIS THOUGHT PROCESS THAT TAKES THE COS BY ITSELF? >>>
and therefore c= pi / 15.
So we have h(t)=a+b*cos((pi/15) * t).
Further we have
h(0)=0:a+b=0
and h(15)=135: a-b = 135.
This implies that
a = -b = 135/2

Found 3 solutions by KMST, Fombitz, rothauserc:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
I read that the London Eye is 135 m tall and has a diameter of 120 m.
That tells me that the bottom is at a height of 15 m.
We need to find , , and to make
a good model for your height above the ground
minutes after you start at the bottom.

WHY COSINE:
We would like to be the start of the ride,
where you probably are at the bottom at ,
and cosine maxes out at ,
where .
That works for ,
with where .
You could use sine, but
would not make the bottom height.
It would be the middle of the ride.

THE AMPLITUDE:
With , the amplitude of is ,
and .
, the amplitude,
is how far a cabin goes from the middle,
and that is half the diameter,
so --> ,
and .

THE HALF-WAY HEIGHT:
is the height of the wheel hub.
,
and you go from to and back
.

THE PERIOD:
Sine and cosine have a period of .
They reach their minimum only once every .
So if you start at the bottom at , and
you return there for the first time at minutes,
at --> .
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
You can use either sine or cosine function to model this since they are the same function shifted in time.
For this problem they chose cosine.
Think of it in terms of periodicity and not trigonometry.
When , the cosine has a value of 1.

.
.
.
When , the cosine should have its minimum value of -1.

.
.
.
is the radius of the London Eye.

Substituting,

So then to calculate , use that minimum value of cosine occurs when

So then putting it all together,

.
..
.
.
.
.
For the illustration, I randomly chose R=100.

Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
The London Eye is a ferris wheel, with h(t)= a + b * cos(c * t)
:
The London Eye has a diameter of 135 meters and passengers board a cabin when theta = 0 degrees at a height of 2 meters above the ground
:
The wheel has a radius of 135 / 2 = 67.5 meters so the height will oscillate about the center with amplitude of 67.5 meters
:
The center of the wheel must be at 67.5 + 2 = 69.5 meters so the midline of the oscillation will be at 69.5 meters
:
The wheel takes 30 minutes to complete one revolution so the height will oscillate with period 30 minutes
:
Since the rider boards at the lowest point, the height will start at the smallest value and increase, this follows the shape of a flipped cosine curve
:
Period is 30, so b = (2 * pi) / 30 = (pi / 15)
:
we could use a transformation of either the sine or cosine function, so
we look for characteristics that would make one function easier than
the other.
:
If we consider the graph of a cosine function that starts at the lowest height value, then
:
we want cosine function because it starts at either the highest or lowest value,
while a sine function starts at the middle value.
:
We know the cosine curve(in this case) was reflected because a standard cosine starts at the highest value, and this graph starts at the lowest value.
:
h(t) = -67.5 * cos( (pi/15) * t ) + 69.5
:
That should get you started to get the answer
:

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