# Design A Rollercoaster

## Grand Opening

This upcoming summer of 2016, the first roller coaster in the Brampton area will open. It is known to be really fast and has a 290 foot drop. The duration of the ride is 100 seconds and some say that, "it looks like a bunch of mathematical equations that were combined by grade 12 students!". Ironically, the ride was designed by student geniuses from Mr. Sharma's class at Castlebrooke Secondary School. The rollercoaster is well known for its name, 'The Beast of the East™". We hope you are brave enough to read our report.

## Final Thoughts-Summary for Plan

We created our rollercoaster by first drawing a rough draft of the rollercoaster on grid paper and we made sure that it included all the required restrictions: it had a duration of 100 seconds; a minimum height of 10 ft and a maximum height of 100 ft. Some difficulties we faced were putting a loop into our graph since we could not go backwards in time. Another difficulty we faced was figuring out a way to have no gaps in our rollercoaster in between each equation. We had to be so precise that we had as much as 7 digits in some of our equations to make sure that our 2 equations aligned. Lastly, the steepness for some of the equations were unrealistic since they almost formed a 90 degree drop. We decided to add simple parabola's with vertical compression so that the drop would have an angular drop. All in all, the conflicts that we encountered got solved!

## Report-Characteristics of the Graph

The height for time graph can be described by using the axis of the graph. The x axis is the time component and y axis is the height component. We put a time restriction of 0 to 100 seconds and put the height restriction at 0 to 300 feet.

The equations used:

Equation #1 (Red Linear function): Y=10{0<x<5.07}

Equation #2 (Blue Linear Function): y=20(x-5.0689654)+10{10<y<300}

Equation #3 (Green Reciprocal Function): y=(10)(x-19.5)+10{10<y<300}{x<24.75}

Function #4 (Orange Reciprocal Function): y=-(10)/(x-30)+10{10<y<300}{24.75<x<29.38023}

Equation #5 (Purple Sinusoidal Functions): y=130sin(0.8x+0.19)+155{29.38023<x<42.9518807}

Equation #6 (Red Quadratic Equation): y=10(x-45.3137)^2+100{10<y<155.7819}

Equation #7 (Blue Quadratic Equation): y=-10(x-50.004)^2+210{155<y<210}

Equation #8 (Green Exponential Function): y=2^{-(x-59.06345)}+50{10<y<155}{x<60}

Equation #9 (Red Quadratic Function): y=2(x-60)^2+50.5224808{60<x<65.8}

Equation #10 (Black Polynomial Function): y=(x-68.98134)^3+150{65.8<x}{y<200}

Equation #11 (Blue Exponential Function): y=-2^(x-82.5)+200{72.665<x}{18.993<y<200}

Equation #12 (Red Logarithmic Function): y=-log(x-89.9999)+11{87<x<100}{18.993>y>10}

## Calculation

When solving for when our rollercoaster reached a certain height, we decided to only solve for the functions that we could visually see reach a height of 250 feet and 12 feet by referring back to the graph on Desmos. For instance, when calculating for when the graph reached 250 feet, we only solved for the functions that we could see reach a height of 250 feet. For example, we could visually see that the quadratic equation used in the period between 43 seconds and 47.68 seconds did not pass the height of 250 ft, thus, we decided not to solve for it since the restrictions restrained it from reaching that height.