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Question 1150529: Building a rollercoaster
Create an equation and graph based off the following
•must cross ground level 4 times
•in 100 feet it must cross ground level
•trip must last maximum of 1 minute
•must start and end in ground level
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you're looking at a sine wave or a cosine wave.
we'll use the sine wave.
the general formula for the sine wave is:
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
you can work in degrees or your can work in radians.
since degrees are easier to understand, we'll work in degrees.
the equation i came up with is:
y = 50 * sin(24 * (x - 90/24)) + 50
the amplitude is the distance from the center line of the graph.
when a is 50, the graph goes 50 units above the center line and 50 units below the center line.
frequency and wavelength are related by the formula:
wavelength = 360 / frequency.
frequency = 360 / wavelength.
we want 4 cycles in 1 minute.
1 minute = 60 seconds.
4 cycles in 60 seconds means 1 full sine wave cycle in 15 seconds.
our wavelength is therefore 15.
the frequency becomes 360 / 15 = 24.
the formula so far is y = 50 * sin(24 * x)
the sine wave starts at the center line of the graph.
in order for it to start at ground level, we need to shift it 90 degrees to the right.
when the frequency is 1, the formula would be y = sin(x - 90)
since the frequency is 24, the formula needs to be y = sin (24 * (x - 90/24))
24 * (x - 90/24) results in 24 * x - 90 which gives us the shift of 90 degrees that we desire.
the formula so far is y = 50 * sin(24 * (x - 90/24))
ground level is asumed to be the x-axis.
if we want the center line of the graph to be 50 units above the x-axis, we need to shift the graph up 50 units.
that happens with d.
the formula becomes y = 50 * sin(24 * (x - 90/24)) + 50.
that's our final equation.
here's the graph.
the coordinate points shown are in (x,y) format.
x is the number of seconds.
y is the height in feet.
when x = 0, the height is 0.
when x = 7.5 seconds, the height is 100 feet.
when x = 15 seconds, the height is back to 0 feet.
the roller coaster touches the ground every 15 seconds for 60 seconds.
it therefore touches the ground 4 times in 60 seconds after it starts.
at (0,0), the height is 0 feet at 0 seconds of travel.
at (7.5,100), the height is 100 feet at 7.5 seconds in travel.
at (15,0), the height is 0 feet at 15 seconds in travel.
this happens 4 times until (60,0), where the height is 0 feet at 60 seconds in travel.
every seconds, the roller coaster has gone up 100 feet and then descended back to ground level 4 times.
this same grph happens every 60 seconds.
here's the same graph going theough 3 sets of 4 cycles each.
first set is 0 to 60.
second set is 60 to 120.
third set is 120 to 180.
each cycle shows the fourth time the roller coaster reaches 100 feet.
every 60 seconds, the roller coaster is back to 0 feet (ground level).
hee's a reference.
https://www.purplemath.com/modules/grphtrig.htm
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