a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift.
the frequency is equal to 360 degrees divided by the period.
since the period is 45 degrees, the frequency is equal to 360 / 35 = 8.
the formula becomes y = a * sin(8 * (x - c)) + d
note that the period is in degrees.
the number of degrees in the period are made the same as the number of seconds for one revolution.
one complete revolution will be made in 45 degrees which is the same as 45 seconds.
the maximum height is 4 meters above the ground and the minimum height is 1 meter above the ground.
the amplitude is equal to 1/2 * (the maximum height minus the minimum height).
that makes it equal to 1/2 * (4 - 1) = 1/2 * 3 = 1.5.
the formula becomes:
y = 1.5 * sin(8 * (x - c)) + d
the horizontal center line of the graph is equal to the highest point on the graph minus the amplitude.
it will also be equal to the lowest point on the graph plus the amplitude.
the highest point is 4.
that minus 1.5 gives you a center line of y = 2.5
the lowest point is 1.
that plus 1.5 give you a center line of y = 2.5.
the horizontal center line on the graph is therefore at y = 2.5
that's the vertical shift.
the formula becomes:
y = 1.5 * sin(8 * (x - c)) + 2.5
the graph will start at the lowest point.
that is at y = 1.
when y = 1, the formula becomes:
1 = 1.5 * sin(8 * (x - c)) + 2.5
subtract 2.5 from both sides of the equation to get:
-1.5 = 1.5 * sin(8 * (x - c))
divide both sides of the equation by 1.5 to get
-1 = sin(8 * (x - c))
solve for 8 * (x - c)) to get:
8 * (x - c)) = arcsin(-1) = -90 degrees.
this is most likely what you calculator will tell you, if you have it set to degrees.
divide both sides of the equation by 8 to get:
x - c = -90/8 = -11.25
since x is 0 when the graph starts, you get:
-c = -11.25
solve for c to get x = 11.25
that would be your horizontal shift.
formula becomes:
y = 1.5 * sin(8 * (x - 11.25)) + 2.5
at 100 seconds, the formula becomes y = 1.5 * sin(8 * (100 - 11.25)) + 2.5 = 2.239527733 meters above the ground.
here's a graph of the first cycle of the ride.
at 100 seconds, the ride has done 2 complete revolutions and then an additional 10 seconds of revolution after that.
the height will be at y = 2.239527733
