A ferris wheel is 20 meters in diameter and boarded from a platform that is 3 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 10 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t)
To find the equation for h = f(t), we need to take into account two things: the height of the ferris wheel above the ground and the time it takes for one full revolution.
Let's start by considering the position of the ferris wheel at the six o'clock position. We know that at this position, the height above the ground is 3 meters. In terms of time, this corresponds to t = 0 minutes.
As the ferris wheel rotates, it completes one full revolution in 10 minutes. Therefore, after t minutes, the wheel will have completed t/10 revolutions.
For each revolution, the ferris wheel goes through a complete cycle of going up, reaching the peak, coming down, and reaching the bottom. This means that in each revolution, the ferris wheel covers a distance equal to its circumference, which can be calculated using the formula C = π * d, where d is the diameter of the wheel.
The diameter of the ferris wheel is given to be 20 meters, so its circumference is C = π * 20 = 20π meters.
Now, the height of a point on the ferris wheel above the ground can be determined by the distance traveled along the circumference in terms of revolutions. For each revolution, the wheel covers 20π meters.
So, the height of the ferris wheel above the ground, h, can be defined as:
h = f(t) = 3 + (t/10) * (20π)
Therefore, the equation for h = f(t) is h = 3 + (2π/5) * t.
This equation gives you the height above the ground at any given time t, in minutes, after the ferris wheel begins to turn.
To write an equation for h = f(t), we need to consider the height of an individual on the ferris wheel at any given time t.
Given that the ferris wheel has a diameter of 20 meters, we can determine that the radius (r) is half of the diameter, which is 10 meters.
Since the loading platform is 3 meters above the ground, a person standing on the platform is at a height of 3 meters at the six o'clock position on the ferris wheel.
As the wheel rotates, it completes one full revolution in 10 minutes. This means that in 10 minutes, the position of the person will be back at the six o'clock position, and the height will have returned to 3 meters.
Since the ferris wheel is a circle, the height of a person on the ferris wheel can be represented by a trigonometric function such as sine or cosine.
The general equation for the height of a person on the ferris wheel at any given time t can be written as:
h = r * cos(theta) + c
Where:
- h is the height above the ground.
- r is the radius of the ferris wheel (10 meters in this case).
- theta is the angle in radians.
- c is the vertical shift, in this case, the initial height of the platform (3 meters).
Since the person is at the six o'clock position initially, the angle theta at time t=0 is 0 radians. And since the person returns to the six o'clock position every 10 minutes, we can determine that the angular velocity of the wheel (omega) is 2π/10 radians per minute.
Therefore, the equation for h = f(t) is:
h = 10 * cos((2π/10) * t) + 3
assuming we take f(0) = 3 because that's where a seat starts, we want f(x) to have a minimum of 3, an amplitude of 10 (the wheel radius) and a period of 10
f(t) = 3 + 10(1-cos(2pi t/10))
since cos has a max at t=0, 1-cos has a min.