Here is the question :

A Ferris wheel is 40 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. How many minutes of the ride are spent higher than 37 meters above the ground?

I'm thinking about the unit circle with this, but I know we also probably need to make a graph because they give us a period other than 2pi. The period would be 4 right? Because it fully rotates in 4 minutes. I don't know what the amplitude would be. Maybe 35 because the maximum is 40 and it is 5 meters above ground.

To solve this problem, we need to understand how the height of the Ferris wheel changes over time.

First, let's determine the equation that represents the height of the Ferris wheel above the ground at any given time. Since the Ferris wheel is 40 meters in diameter, the radius would be half of that, which is 20 meters.

We can derive the equation for the height of the Ferris wheel using the sine function, as it represents the periodic motion of the Ferris wheel.

The general equation for the height (h) of the Ferris wheel as a function of time (t) is:

h(t) = A * sin(2π * t / T) + C

where A is the amplitude, T is the period, and C is the vertical shift (height above the ground).

From the information given, the amplitude (A) is the maximum height of the Ferris wheel above the ground, which is the radius (20 meters) minus the height of the loading platform above the ground (5 meters). So, the amplitude is 20 - 5 = 15 meters.

The period (T) is the time taken for the Ferris wheel to complete one full revolution, which is given as 4 minutes.

The vertical shift (C) is the initial height above the ground when the wheel is in the six o'clock position. Since the loading platform is 5 meters above the ground, C is 5 meters.

So, the equation that represents the height (h) of the Ferris wheel at any given time (t) is:

h(t) = 15 * sin(2π * t / 4) + 5

To find the number of minutes of the ride spent higher than 37 meters above the ground, we need to determine the time range during which the height of the Ferris wheel is greater than 37 meters.

We can set up the inequality:

h(t) > 37

Simplifying the inequality, we get:

15 * sin(2π * t / 4) + 5 > 37

Subtracting 5 from both sides:

15 * sin(2π * t / 4) > 32

Dividing both sides by 15:

sin(2π * t / 4) > 32/15

Using inverse sine (or arcsin) to solve for t, we can write:

2π * t / 4 > arcsin(32/15)

Simplifying, we get:

π * t / 2 > arcsin(32/15)

Multiplying both sides by 2/π:

t > 2 * arcsin(32/15) / π

Now, you can use a calculator to find the value of arcsin(32/15) and then substitute it into the inequality to find the value of t. This will give you the number of minutes of the ride spent higher than 37 meters above the ground.

To answer the question, we need to understand the height of the Ferris wheel at different positions during its rotation.

Let's start by visualizing the situation described. The Ferris wheel has a diameter of 40 meters, so the radius is half of that, which is 20 meters. The Ferris wheel is boarded from a platform that is 5 meters above the ground. Therefore, the center of the Ferris wheel is located at a height of 5 meters + 20 meters = 25 meters above the ground.

Next, let's consider the rotation of the Ferris wheel. We are told that it completes one full revolution in 4 minutes. This means that it takes 4 minutes to go from the starting position (six o'clock) back to the same position.

To find out the height of the Ferris wheel at any given time, we can use the unit circle. In the unit circle, the radius is 1, but in our case, the radius of the Ferris wheel is 20 meters. To find the height at any given angle, we need to multiply the sine value of that angle by the radius of the Ferris wheel (20 meters).

Since the starting position (six o'clock) is level with the loading platform, which is 5 meters above the ground, the height formula becomes:

height = 25 + 20*sin(angle)

Now, we need to find the angles at which the height is higher than 37 meters.

Let's calculate the highest point on the Ferris wheel by finding the maximum value of the sine function. The maximum value of the sine function is 1, so the maximum height occurs at sin(angle) = 1. In this case, the height would be 25 + 20*1 = 45 meters.

To find the angles corresponding to a height higher than 37 meters, we need to solve the equation:

37 < 25 + 20*sin(angle)

Subtracting 25 from both sides, we get:

12 < 20*sin(angle)

Now divide both sides by 20:

12/20 < sin(angle)

0.6 < sin(angle)

To find the angles at which sin(angle) is greater than 0.6, we can use the inverse sine function (also known as arcsine or sin^(-1)).

The arcsine function returns the angle whose sine is a given value. In this case, we want angles where sin(angle) is greater than 0.6. So, we need to find:

angle > arcsin(0.6)

Using a calculator or a table of trigonometric values, we can find that arcsin(0.6) is approximately 36.87 degrees.

Considering the unit circle, this angle corresponds to about 2/3 of a complete revolution. Since a complete revolution takes 4 minutes, 2/3 of a revolution would take:

4 minutes * 2/3 = 8/3 minutes ≈ 2.67 minutes

Therefore, the number of minutes spent higher than 37 meters above the ground is approximately 2.67 minutes.

radius = 20

height of center = 25
height = 25 + 20 sin [2 pi t/4 - pi/2}

I use 2 pi t/4 because that gives 0 at t = 0 and a full revolution, 2 pi, at t = 4.
I put in the phase of -pi/2 so that at t = 0 we have 25 +20 sin -pi/2
= 25 - 20 = 5 meters, the starting point at t = 0
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Now, all of that said, it is not necessary
we need to know what fraction of the circumference is above 37 meters or how much of the circle is more than 37-5 = 32 meters above bottom point of circle
which is 12 meters above center.
T is angle from horizontal through center to 12 feet above center
sin T = 12/20
T = 37 degrees approximately
so out of the 180 degree upward swing, the rider spends 180 - (37+90) = 53 degrees above 37 meters on the rise
or a total of 106 degrees including the start of the fall
the wheel spends 4 minutes going from bottom to top so
time above = (106/360)4 = 1.18 minutes