The Labyrinth

By: Sumiksha, Karishma, Amandeep, and Deepinder

Description

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About the Labyrinth

The Labyrinth was established in 1988 and recently innovated into a modern roller coaster, with high inclined drops and a longer roller coaster experience. The Labyrinth is found only at Six Flags amusement park in Jackson, New Jersey. Famous engineers such as, Carl Johnson and Austin Davis have worked together to create the breath taking structure of The Labyrinth.

General Description

The roller coaster has a maximum height of 300 feet and a minimum height of 10 feet. After 85 seconds have passed by the roller coaster reaches it’s maximum height of 300 feet. The first 13 seconds of the ride the roller coaster is at it’s minimum height of 10 feet. From 14-19 seconds it goes back to it’s minimum height and again at 23-28 seconds. At the end of the ride the coaster reaches its minimum from 90 seconds to the end. From 32 seconds to 49 seconds the roller coaster is at the highest for the longest period of time compared to the rest of the ride.


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Equations

Linear:

1.

y= -100(x-87.96) +273.92 {y<273.92} {y>20}

Quadratic:

2.

y= -12(x-21.5)² +36 {x>10} {y>10}


3.

y= -13 (x-68)² +148 {x>65.35} {x<69.69}


4.

y= -15(x-9.6)² +238.4 {x>7.75} {x<13.43}

Polynomial

5.

y=1/7 (x-40)(x-65)(x-60)+50 {y>1} {y<300}{ y>17.5} {x<65.358} {x>45.96}


6.

y= (x-78)³ +144 {x>76.804}{y<300}{y<272.69}


7.

y= -(x-85.7)⁴ +300 {y>272.69} {x<87.9601}


8.

y= 1/0.5 (x-33)³+270 {x<34.9} {x>27.98}

Rational

9.

y= 1/(x-90.38)+12 {x>90.5} {y>0} {x<100}

Sinusodial

10.

y= -9sin(1/2x-0.29π)+275.9 {x<45.96} {x>35}


11.

y= -1/0.3sin(x-3π)+109 {x?69.69}{x<76.804}

Exponential

12.

y= 2^ 2(x-4.02) +10 {y<187.053} {x>0}


13.

y= 2^ -2^(x-15)+10 {y<54.63} {x<20} {x>13.43}

Logarithimic

14.

y=-1/2log(-x +28) +10.5 {x>22.97}

Characteristics

y= -100(x-87.96) +273.92 {y<273.92} {y>20}

-slope of a 100

-reflection in the x-axis

-horizontal translation 87.96 units to the right

-vertical translation 273.92 units upward

y= -12(x-21.5)² +36 {x>10} {y>10}

- reflection in the x-axis

-vertical stretch by a factor of 12

-horizontal translation 21.5 units to the right

-vertical translation 36 units upward

-even degree function

y= -13 (x-68)² +148 {x>65.35} {x<69.69}

-reflection in the x-axis

-vertical stretch by a factor of 13

-horizontal translation 68 units to the right

-vertical translation 148 units upward

-even degree

y= -15(x-9.6)² +238.4 {x>7.75} {x<13.43}

-reflection in the x-axis

-vertical stretch by a factor of 15

-horizontal translation 9.6 units to the right

-vertical translation 238.4 units upward

-even degree

y=1/7 (x-40)(x-65)(x-60)+50 {y>1} {y<300}{ y>17.5} {x<65.358} {x>45.96}

-horizontal compression by a factor of 1/7

- zeroes: 40, 65, 60

-vertical translation 50 units upwards

-odd degree function

y= (x-78)³ +144 {x>76.804}{y<300}{y<272.69}
-horizontal transition 78 units to the right

-vertical transition 144 units up

-odd degree function

y= -(x-85.7)⁴ +300 {y>272.69} {x<87.9601}

-reflection in the x-axis
-horizontal shift 85.7 units to the right
-vertical transition 300 units up

-even degree

y= 1/0.5 (x-33)³+270 {x<34.9} {x>27.98}
-horizontal compression by a factor of 1/0.5
-horizontal transition 33 units to the right
-vertical transition 270 units up

-odd degree

y= -9sin(1/2x-0.29π)+275.9 {x<45.96} {x>35}
-reflection in the x-axis
-vertical stretch by a factor of 9
-horizontally compressed by a factor of 1/2
-horizontal translation to the right by 0.29π

-vertical translation 275.9 units up

y= -1/0.3sin(x-3π)+109 {x?69.69}{x<76.804}

-reflection in the x-axis

-horizontal compression by a factor of 1/0.3

-horizontal translation to the right by 3π

-vertical translation up by 109 units

y= 2^ 2(x-4.02) +10 {y<187.053} {x>0}

-vertical stretch by a factor of 2

-horizontal compression by a factor of 1/2

-horizontal translation to the right by 4.02 units

-vertical translation 10 units up

y= 2^ -2^(x-15)+10 {y<54.63} {x<20} {x>13.43}

-vertical stretch by a factor of 2

-horizontal reflection

-horizontal compression by a factor of 1/2

-horizontal translation to the right by 15 units

-vertical translation 10 units up

y=-1/2log(-x +28) +10.5 {x>22.97}

-reflection in the x-axis

-horizontal compression by a factor of 1/2

-reflection in the y-axis

-horizontal translation to the left by 28 units

-vertical translation up by 10.5 units

y= 1/(x-90.38)+12 {x>90.5} {y>0} {x<100}

-vertical stretch by a factor of 1

-horizontal translation to the right 90.38 units

-vertical translation upwards 12 units

Calculations

Solving

250 ft

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12 ft

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Calculating Average Rate of Change

10 to 15 seconds

AROC - -45ft/s

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50 to 60 seconds

AROC- -21.4 ft/s

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Calculating Instantaneous Rate of Change

35 seconds

IROC- 3ft/s

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35.001 Seconds
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Summary

The roller coaster assignment consisted for each group member to work together and brainstorm possible ideas for a roller coaster. We wanted to create a roller coaster that consisted of different equations that had a unique structure. Once we came up with possible quadratic, cubic, quartic and other functions, a rough structure was modeled on desmos. The equations were formed as a rough model on paper, but once we added them to desmos they were changed. One of the biggest difficulty was making the different functions so that they line up with the other functions to create a smooth roller coaster. For example, some functions needed to be moved down so there would be no spaces between surrounding functions. As a group we decided that stretches and compression were optional, so we added them once the function was graphed on desmos. This was a good decision because it helped with timing. If we applied a stretch on the function, it caused it to be narrow such that it would cover less timing. Likewise the same thing applies to functions that have a compression, this made the function wider allowing it to cover more time. Another difficulty our group faced was making restrictions. In order to make the entire roller coaster and every equation connect the restriction had to be really precise. The more we tried to connect it and make it flow the more significant digits we had in our restrictions. To conclude working on this assignment has helped each group member utilize the skills that were taught in grade 12 advanced functions. Most of these skills were applied when equations needed to be solved for certain times and heights. When solving equations skills likes, synthetic division, factoring and graphing.