y = a * sin(b * (x - c)) + d
a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift.
on a graph, the 3 o'clock position is parallel to, or on, the x-axis, due east of the y-axis.
the counterclockwise direction creates an angle whose vertex is at the origin and whose one side is the line parallel to the x-axis and to the right of the y-axis, and whose other side is the line that forms the other side of the angle.
the ferris wheel has a 40 foot radius.
the horizontal center line of the circle formed is 44 feet above the ground.
40 feet is the radius of the circle.
this means that the bottom of the circle is 4 feet above the ground.
the ground is represented by the x-axis.
the equation for the ferris wheel is:
y = a * sin( b * (x - c)) + d
a = 40
c = 0
d = 44
the sine wave graph will have the horizontal center line at y = 44.
that's the d in the equation.
there sill be no horizontal shift, so c will be equal to 0 and will not appear in the equation if it is not desired to show it as 0.
the amplitude is the radius of the circle.
because the horizontal cemter is y = 44, the amplitude will go from y = 44 to y = 4.
the frequency is represented b.
the frequency is equal to the 2 * pi divided by the period of the sine wave.
the ferris wheel rotate in a counter clockwise direction at the constant speed of 2 radians per second.
since the circumference of the ferris wheel is 2 * pi radians, and the ferris rotates at the constant speed of 2 radians per second, then it will take 2 * pi / 2 = pi radians for one complete revolution.
therefore, the period of the sine wave will be pi radians.
the frequency is equal to 2 * pi divided by the period.
that makles the frequency equal to 2 * pi / pi = 2.
the values of the variables for the sine wave function are now:
a = 40
b = 2
c = 0
d = 44
the sine wave function becomes:
y = 40 * sin(2 * (x - 0)) + 44, which becomes:
y = 40 * sin(2 * x) + 40
here's that that looks like on a graph.
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the graph uses x in instead of t, but they both mean the same thing, i.e. minutes from when the ferris wheel starts rotating from the 3 o'clock position.
answers to your questions to the best of my knowledge would be:
a) Write an expression (in terms of t) to represent the number of radians Jordan has swept out from the 3-o'clock position since the ride started.
y = 40 * sin( 2 * t) + 44
t = 0 minutes is when the ferris wheel starts
the number of complete revolutions of the ferris wheel will be every pi minutes
that would be equal to every 3.141592654..... minutes.
b) How long does it take for Jordan to complete one full revolution (rotation)?
it will take him 3.141592654 minutes (equals pi) for one complete revolution of the ferris wheel if there are no stops or pauses between when it starts and when it completes one revolution.
c) Write an expression (in terms of t) to represent Jordan's height above the center of the Ferris wheel (in feet).
the equation is y = 40 * sin(2 * x) + 44.
this gives the center line of the ferris wheel at 44 feet above the ground.
take that away and you'll have jordan's height above the center line of the ferris wheel.
he will fluctuate between 40 feet above the center line of the ferris and 40 feet below the center line of the ferris wheel.
so that equation would be y = 40 * sin(2 * x)
d) Write a function f that determines Jordan's height above the ground (in feet) in terms of t.
that function is the original equation of y = 40 * sin(2 * x) + 44.
here's some graphs that hopefully show you what i mean.
the first show you the sine wave.
the last is a sketch i made of the ferris wheel and it's position on the graph.
i don't have time to explain any further, but if you have any questions or concerns, contact me through algebra.com and i'll explain further as much as i can.