Question 891520: I need help with this word problem:
Niagara Skywheel has a diameter of 50.5m and rotates 1 revolution in 2 minutes. the lowest point of the wheel is 2.5m off the ground.
I need to find the sine equation that models the height of a rider over time on the Skywheel.
any extra explanation is wonderful, and if there is a way to do this without using pi, I would love to know :)
Thank you for any help you can give!
Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! Skywheel has a diameter of 50.5 meters.
It rotates 1 complete revolution in 2 minutes.
The lowest point of the wheel is 2.5 meters off the ground.
You want to find the sine equatgion that models the height of a rider over time.
You want to know whether you can do this without pi.
The answer to this question is that you can, but there's no reason for you to want to unless you like to write a lot of numbers when you need to use pi.
The reason you need to use pi is that the skywheel forms a circle and the circumference of a circle is given by the equation C = 2*pi*r, where r is the radius and pi is a constant that is equal to 3.141592654.
When you use pi, you are using the constant of 3.141592654.
If you don't want to use pi, then you have to use the constant instead, since pi is an integral part of the formula.
It's a lot easier to use pi to represent the constant than to have to write down all those numbers every time.
Use pi and then translate pi to 3.141592654 when you're done.
If you're using a calculator, the calculator will automatically replace pi with 3.141592654 for you.
Back to your problem.
The diameter of the skywheel is equal to 50.5 meters.
this means the radius of the skywheel is equal to 25.25 meters.
This means the circumference of the skywheel is equal to 2 * pi * r which is equal to 2 * pi * 25.25 which is equal to 50.5 * pi.
The circumference of a circle is also equal to 360 degrees.
This means that 50.5 * pi = 360 degree.
Since the skywheel makes 1 revolution in 2 minutes, this means that the skywheel rotates 360 degrees in 2 minutes.
If you look at the picture of a circle and you relate that circle to the skywheel, you will see that the center of the circle will be the center of the skywheel.
You will also see that the low point of the circle is when the angle is equal to 270 degrees and the high point of the circle is when the angle is equal to 90 degrees.
you want the revolution to start at the low point.
the formula for the circle is y = sin(x).
x is the number of degrees.
the equation you want to start with is y = sin(x).
If you graph that equation, you will see that it starts at sin(x) = 0, goes up to sin(90) = 1, goes down to sin(180) = 0, goes down to sin(270) = -1 and ends up at sin(360) = 0
the low point is when sin(x) = -1.
that's when x = 270 degrees.
you want the wheel to be at the low point when x = 0, so you have to shift the equation to the left by 270 degrees.
your equation of y = sin(x) becomes y = sin(x + 270)
now when x = 0, sin(x + 270) will be equal to sin(270) which will be equal to -1.
the graph of y = sin(x + 270) is shown below:
now your circle has a radius of 25.25 meters, so you need to add that into the equation and you want the center of your equation to be 25.25 meters above the ground and you want the bottom travel of the circle to be 2.5 meters above the around.
your equation will become:
y = 25.25 * sin(x + 275) + 27.75
the 25.25 becomes the amplitude of the sine wave which will fluctuate between + 25.25 and -25.25.
the 27.25 raises the whole equation so that the center of rotation is now at 27.25 meters instead of at 0 meters.
the low point now becomes 27.25 - 25.25 = 2.5.
the high point now becomes 27.25 + 25.25 = 52.5
the graph of y = 25.25 * sin(x + 275) + 27.75 is shown below:
now you have the equation where you want it and the only thing left to do is to relate the revolution of the skywheeel to minutes instead of degrees.
the x-axis is now in degrees.
you want to convert it to minutes.
since 360 degrees is equivalent to 2 minutes, you want the x-axis to read 2 minutes where it used to read 360 degrees.
the skywheel will make 1 revolution in 2 minutes rather than in 360 degrees.
to do that the frequency of the skywheel has to be multiplied by 180 because it will now make 180 revolutions in 360 degrees rather than 1 revolution in 360 degrees.
the shifting of the skywheel will change from 275 degrees to 275 / 180 = 1.5 minutes.
your equation will become:
y = 25.25 * sin(180 * (x + 1.5)) + 27.75
now you're done.
the graph of y = 25.25 * sin(180 * (x + 1.5)) + 27.75 is shown below:
a picture of your skywheel is shown below.
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