A ferris wheel has a radius of 20m and the bottom is 2m above the ground. The wheel makes one rev in 40 sec. If the rider gets on at 30^0 on the left of the lowest point going downwards, wirte the sine function to show this movement
the axle is at height 2+20 and the amplitude is 20, so we can start with
f(t) = 22+20sin(t)
we want a period of 40 seconds, so 2π/k=40, making k=π/20
f(t) = 22+20sin(π/20 t)
-cos(x) achieves its minimum at x=0, so since we want our minimum at 30°, which is 1/12 of a period, or at t=10/3
f(t) = 22-20cos(π/20 (t-10/3))
Now, cos(x) = sin(π/2-x), so you can massage that into the sine curve
f(t) = 22-20sin(π/20 (t+20/3))
the wolframalpha page confirms that this has a minimum at t=10/3, or 1/12 of a period. That is, 30° after starting.
http://www.wolframalpha.com/input/?i=22-20sin(%CF%80%2F20+(20%2F3%2Bt))
To write the sine function for the given movement of the rider on the ferris wheel, we need to consider the angle and the height of the rider at each point.
Let's assume the ground level as the reference point (y = 0), and the positive y-axis pointing upwards.
Since the rider gets on at 30° left of the lowest point (going downwards), we can consider this as the starting position (0°).
At the lowest point, the height of the rider would be 2 meters above the ground.
As the wheel rotates counterclockwise, the angle increases. After completing one revolution (360°), the rider would be back at the same position as the starting point.
Now, let's consider the height of the rider as a function of the angle.
Given:
Radius of the ferris wheel (r) = 20 meters
The rider gets on at 30° left of the lowest point (going downwards), which is the starting position.
The lowest point is 2 meters above the ground (initial height).
To express the height (h) of the rider as a function of the angle (θ), we can use the sine function.
The general equation for the sine function is:
sin(θ) = opposite/hypotenuse
In this case, the rider's height (h) is the opposite side, and the radius (r) of the ferris wheel is the hypotenuse.
sin(θ) = opposite/hypotenuse
sin(θ) = h/r
Substituting the given values:
sin(θ) = h/20
Since the rider gets on at 30° left of the lowest point (going downwards), we can represent this location as θ = -30° (negative angle since it is left of the starting position).
Therefore, the sine function to represent the movement of the rider is:
sin(-30°) = h/20
Simplifying the equation:
h/20 = sin(-30°)
This equation represents the height of the rider as a function of the angle during the movement on the ferris wheel.
To write the sine function to show the movement of the rider on the Ferris wheel, we need to consider the position of the rider at any given time.
Let's define the angle θ as the angle between the vertical axis and a line connecting the center of the wheel to the rider. At the starting position, θ is 30 degrees, going downwards.
The sine function relates the angle θ to the vertical displacement of the rider from the lowest point on the wheel. Since the bottom of the wheel is 2m above the ground, we need to subtract this value from the vertical position.
Since the radius of the Ferris wheel is 20m, the maximum vertical displacement from the lowest point is 20m as well. So, the amplitude of the sine function is 20m.
The period of the sine function is the time it takes for the Ferris wheel to complete one revolution. Given that it takes 40 seconds for one revolution, the period is 40 seconds.
Now we can write the sine function to represent the rider's vertical position as a function of time t:
h(t) = -20sin((2π/40)t - π/6) + 2
In this equation, t represents time in seconds, and h(t) represents the height of the rider above the ground at time t. The negative sign accounts for the downward direction of the initial angle, and the π/6 phase shift adjusts for the starting position at 30 degrees.
Remember to convert the angle to radians when using it in trigonometric functions.