Question 1118610: The amusement park has a Ferris wheel with a radius of 4 m and it takes 1 minute to make one revolution. The lowest carriage is 1 m above the ground. Consider the height of a specific carriage as the wheel rotates
a. What are the independent and dependent variables?
b. Construct a table of values.
c. What is the mathematical model for this situation?
d. Graph the function.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ferris wheel forms a circle with radius 4 meters.
a carriage on the ferris wheel is positioned on the circumference of this circle.
you can use the equation of a sine wave to simulate the height of the carriage from the ground at different points in the revolution of the carriage around the circumference of the circle.
the equation of a sine wave is y = a * sine (b * (x-c)) + d
a is the amplitude of the sine wave.
b is the frequency of the sine wave
c is the horizontal displacement of the sine wave.
d is the vertical displacement of the sine wave.
the period of the sine wave is equal to 360 degrees when b is equal to 1.
without changing that, b = 1 and disappears from the equation which now becomes y = a * sine ((x - c)) + d.
the amplitude of the sine wave is a.
with a radius of 4 meters, the amplitude becomes 4 and the equation of the sine wave becomes y = 4 * sine ((x-c)) + d.
with no horizontal displacement, c = 0 and the equation of the sine wave becomes y = 4 * sine (x) + d.
with no vertical displacement, d = 0 and the equation of the sine wave becomes y = 4 * sine (x).
this makes the center line of the equation equal to 0 and the equation will have a high and low peak of y = 4 and -4 and the period of the wine wave will be 360 degrees.
the equation would look like this:
you can see that y fluctuates from 4 to -4 and the center line is at y = 0.
to simulate the ferris wheel, we want the bottom of the sine wave to be 1 meter above y = 0, which represents ground level.
therefore, we have to raise the center line from 0 to 5.
that will make the sine wave have a high value of y = 9 and a low value of y = 1.
we do this by making d = 5.
that raises the center line 5 units above y = 0.
the equation becomes y = 4 * sine (x) + 5.
it looks like this:
you can see that that the value of y now goes from a high point of 9 to a low point of 1 with a center line of y 5.
the ground level is still at y = 0 which means that the low point of the sine wave is 1 meter above the ground.
if we want the carriage of the sine wave to be at ground level when x = 0 degrees, then we have to shift the sine wave 90 degrees to the right.
we do that with c.
the sine wave equation becomes y = 4 * sine (x - 90) + 5 which shifts the sine wave 90 degrees to the right.
since the sine wave was at the bottom peak at x = -90 degrees, by making the equation equal to sine (x-90) rather than sine (x), when x = 0, we get the sine (-90) which is at the bottom peak of the sine wave.
y = 4 * sine (x - 90) + 5 looks like this:
it looks like we're done, except we may want the value of x to represent seconds rather than degrees.
since we want the sine wave to make one complete revolution in 60 seconds, then we would want the period of the sine wave to be equal to 60 degrees rather than 360 degrees.
in that way, we have the sine wave making one complete revolution in 60 seconds which is equal to 1 minute.
the formula for period is period = 360 degrees divided by frequency.
since we want the period to be 60 degrees, the formula becomes 60 = 360 / frequency
we solve for frequency to get frequency = 360 / 6 = 6.
since b is the frequency, the sine wave formula now becomes y = 4 * sine (6 * (x-90)) + 9
we need, however, to adjust the horizontal displacement as well.
when the period was 360 degrees, the horizontal displacement was 90 degrees.
we divided 360 by 6 to get a period of 60.
we need to divide the horiontal displacement by 6 to get it in synch with the new period.
that would make the horizontal displacement 90 / 6 = 15 degrees.
the equation now looks like y = 4 * sine(6 * (x - 15)) + 9.
that equation looks like this on the graph:
we now have what we want.
at 0 seconds, the carriage is at 1 meters above the ground, represented by the value of x.
it makes a complete revolution in 60 seconds, represented by the value of x.
it is at it's highest height above the ground in 30 seconds, represented by the value of y = 9 when the value of x = 30.
the table of values was created using the excel spreadsheet program.
the height of the carriage is represented by the value of y.
make a table of values from x = 0 to x = 60.
calculate the value of y for each value of x using the equation of y = 4 * sine (6 * (x - 15)) + 9.
the table looks like this:
you can double check the values in the table by using the formula of y = 4 * sine (6 * (x-15)) + 5.
just replace x with any value from 0 to 60 and your answer should be the same as in the table.
make sure your calculator is set to degrees before doing so.
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