A carnival ferris wheel that has a diameter of 18m, goes 4 seconds every revolution, there are 5 revolutions, also the ferris wheel is 2m above the ground
Cosine and Sine equation needed
you don't say at what point the wheel starts. Let's say that at t=0 the point of interest at the bottom of the rotation.
Since cos(t) starts at the top, we know right away that the function will look like
y = -cos(kt)
The diameter is 18, so the radius is 9, and we now have
y = -9cos(t)
But, the starting chair is 2m above the ground, rather than 9m below, so
y = -9cos(t) + 11
The period is for seconds, so
y = -9cos(π/2 t) + 11
No sure what to do with the 5 rotations thing. That means the ride lasts 20 seconds. What a rip-off!
As for the sine function, you know that cos(x) = sin(π/2 - x) so fit that in.
Or, consider that an unshifted sine function would start when the loading chair was at axle height.
To solve this problem, we can use the cosine and sine functions to find the height and horizontal displacement of the ferris wheel at a given time.
Let's start with the cosine function. The general equation for the cosine function is:
cos(θ) = adjacent / hypotenuse
In this case, we can consider the angle θ as the angle through which the ferris wheel has turned. The adjacent side represents the horizontal displacement from the center of the ferris wheel, which is equal to the radius, and the hypotenuse represents the diagonal distance from the center to the position of the ferris wheel.
Since we know the diameter of the ferris wheel is 18m, the radius (adjacent) will be half of that, which is 9m. The diagonal distance (hypotenuse) is the sum of the radius and the height above the ground. Given that the ferris wheel is 2m above the ground, the hypotenuse is 9m + 2m = 11m.
So, the equation becomes:
cos(θ) = 9 / 11
Now, let's move on to the sine function. The general equation for the sine function is:
sin(θ) = opposite / hypotenuse
In this case, the opposite side represents the vertical displacement from the center of the ferris wheel, which is equal to the height above the ground.
Again, using the height above the ground of 2m, the equation becomes:
sin(θ) = 2 / 11
By using these equations, we can find the height and horizontal displacement of the ferris wheel for any given angle or time.
Note that you mentioned the ferris wheel goes for 4 seconds per revolution and there are 5 revolutions. To calculate the time it takes for each revolution, you can divide the total time by the number of revolutions. In this case, the total time would be 5 revolutions multiplied by 4 seconds per revolution, which equals 20 seconds. So, each revolution takes 20 seconds / 5 revolutions = 4 seconds.
I hope this explanation helps you understand how to use the cosine and sine equations to find the height and horizontal displacement of the ferris wheel at any given time during its rotation.