# THE MIGHTY MILLENNIUM FORCE

## How did we created our roller coaster

Creating the rollercoaster was no easy task, as it required an enormous amount of time and patience to complete it. We had initially started off by planning on how the roller-coaster will look like. We decided to model it after the letter M to show our passion for math. We then compiled a list of all the parent functions and using that knowledge we did a rough sketch of how we wanted our final graph to look like. We had to take many factors into consideration, such as the maximum and minimum values, the degree for some functions for example, cubic functions, as well as the time limit for the entire graph. Next, we tried to apply transformations to the equations of the functions and restrictions in the domain and range of their equations. In order for us to correctly visualize how our graph was turning out we started using the graphing software, Desmos. In the end, we had finally come up with all of our equations, with the restrictions and it had seemed to connect perfectly, but after closer inspection (by zooming in) we had realized that some of the equations were not actually attached to each other, or they were over lapping each other. With this in mind, we went back to Desmos and started tweaking the equations in order for them to connect properly by changing the restriction.

## General description of function of height vs. time

In our graph, time is an independent variable and height is a dependent variable. The graph time is constantly increasing, but the height is fluctuating. Up to the point of (0, 10) to (4.6, 300) the graph’s slope increased exponentially. Then we added three different cosine functions to our graph making the height of the increase and decrease. We made the graph decrease in slope for about 26 seconds starting from the fifth second to 31, because this will be extremely scary and will add thrills for the foolish daredevils that believed that they could conquer the Mighty Millennium Force. We allowed the riders to catch their breaths. During the next 40 seconds when the height of the roller-coaster had minimal changes, allowed the riders to regret coming aboard this ride, and allowed them ample time to write their will and say their dying wishes to the person beside them. We made the roller-coaster increase in height gradually giving our victims more time to regret listening to the person that told them to go on this ride. The last five seconds will mostly like be the final seconds of your lifetime. You will encounter a very steep curve, where our trained statistician has calculated the survival rate will be 0.00001%. This curve is very steeply going downwards. We tried to make this part resemble the first drop of Levitation at Canada’s Wonderland. If you are reading beyond this point, after you have ridden this roller-coaster, you are truly one in a million, CONGRATULATIONS ON CONQUERING THE MIGHTY MILLENNIUM FORCE.

## Equations and their restrictions

1. Y = 3x^3 + 10 {x ≥ 0} {{x ≤ 4.59}

2. Y = 4cos [pi /1.5 (x - 7)] -10 (x - 7) + 274.7 {x ≥4.59} {x ≤ 10.84}

3. Y = 7cos [2(x - 10.05464)] + 236.55 {x ≥ 10.84} {x ≤ 16.36}

4. Y = 4cos [pi/1.5 (x - 7)]-10 (x - 7) + 334.2 {x ≥ 16.3582} {x ≤ 24.98}

5. Y = -2(x-30) (x + 0.005) + 159 {x ≥ 24.98} {x ≤ 29.31}

6. Y = -x^2 + 1000 {x ≥ 29.31} {x ≤ 30.651}

7. Y = log 5 + 59.79 {x ≥ 30.65} {x ≤ 35.651}

8. Y = (2x / x + 1) + 58.55 {x ≥ 35.64} {x ≤ 39.9}

9. Y = 7cos [2(x - 11.5464)] + 53.5 {x ≥ 39.9} {x ≤ 43.485}

10.Y = -x + 100.55 {x ≥ 43.485} {x ≤ 50}

11.Y = x + 0.55 {x ≥ 50} {x ≤ 56.07}

12.Y = 7sin [2(x - 55.84)] + 53.55 {x ≥ 56.07} {x ≤ 59.767}

13.Y = (2x / x + 1) + 58.55 {x ≥ 59.767} {x ≤ 64.027}

14.Y = log 5 + 59.79 {x ≥ 64.027} {x ≤ 69.68}

15.Y = x^2- 4800 {x ≥ 69.718} {x ≤ 70.29}

16.Y = -2(-x+68.5) (x + 0.005) + 159 {x ≥ 70.29} {x ≤ 75.95}

17.Y = 4 cos [pi / 1.5 (x - 7)] + 10 (x - 7) - 535 {x ≥ 75.95} {x ≤ 83.93}

18.Y = 7sin [2(x - 10.05464)] + 232.55 {x ≥ 83.93} {x ≤ 89.38}

19.Y = 4sin [pi / 1.5 (x + 7)] + 10 (x - 7) – 587 {x ≥ 89.38} {x ≤ 95.41}

20.Y = -3(x - 100)^3 + 10 {x ≥ 95.41} {x ≤ 100}

## Calculations

12 Feet

Y = 3X^3 + 10

12 = 3X^3 + 10

2 = 3X^3

0.6667 = X^3

0.8735 = X

Y = -3(X - 100)^3 + 10

12 = -3(X - 100)^3 + 10

2 = -3(X - 100)^3

-0.6667= (X - 100)^3

-0.8735 = X – 100

99.1264 = X

The exact time when the roller-coaster reaches the height of 12 feet is at 0.87 and 99.13 seconds.

Y = 3X^3 + 10

250 = 3X^3 + 10

240 = 3X^3

80 = X^3

4.3088 = X

Y = -3(x - 100)^3 + 10

250 = Y = -3(x - 100)^3 + 10

240 = -3(x - 100)^3

-80 = (x - 100)^3

-4.3088 = (x - 100)

95.6911 = x

The exact time when the roller-coaster reaches the height of 250 feet is at 4.31, 9.97, 91.07, and 95.69 seconds.

The average rate of change from 10 to 15 seconds is:

Y value for 10 seconds

Y = 4cos [pi /1.5 (x - 7)] -10 (x - 7) + 274.7

Y = 4cos [pi /1.5 (10 - 7)] -10 (10 - 7) + 274.7

Y = 4cos [pi /1.5 (3)] -10 (3) + 274.7

Y = 4cos [pi /4.5] -30 + 274.7

Y = 3.999 +224.7

Y = 248.699

15 seconds

Y = 7cos [2(x - 10.05464)] + 236.55

Y = 7cos [2(15 - 10.05464)] + 236.55

Y = 7cos [2(4.94536)] + 236.55

Y = 230.296

(10, 249.7) and (15, 230.3)

Aroc = Y2 - Y1 / X2 - X1

Aroc = 248.7 - 230.3 / 10 - 15

Aroc = 18.4 / -5

Aroc = 3.68 Feet / Second

From the point 10, 249.7 and 15, 230.3 the average rate of change is - 3.68 feet per second

The average rate of change from 50 to 60 seconds

Y value for 50 seconds

Y = X + 0.55

Y = 50 +0.55

Y = 50.55

Y value for 60 seconds

Y = (2X / X + 1) + 58.55

Y = (2(60) / 60 + 1) + 58.55

Y = (120 / 61) + 58.55

Y = 1.9672 +58.55

Y = 60.5172

(50, 50.55) and (60, 60.5172)

AROC = Y2 - Y1 / X2 - X1

AROC = 60.5172 - 50.55 / 60 - 50

AROC = 9.9672 / 10

AROC = 0.99672 Feet / Second

From the point 50, 50.55 and 60, 60.52, the average rate of change is 0.997 feet per seconds.

The Instantaneous rate of change from at 35 seconds

Y = log 5 + 59.79

(35, 60.49) and (35.001, 60.489)

IROC = Y2 - Y1 / X2 - X1

IROC = 60.489 – 60.49/ 35.001 - 35

IROC = - 0.001 / 0.001

IROC = -1 Feet / Second

The Instantaneous rate of change is -1feet per seconds at 35 seconds.