# Design A Rollercoaster

### The SkyRunner

By: Chanjot, Chenique, Khyati, Yesha.

## Math Summative

In the last year of grade 12 mathematics (Advanced Functions), we were ask to design a roller coaster that is 10ft above the ground and goes a maximum height of 300ft as our summative.

## Summary of Plan

While working on this projects our group faced many difficulties but we were able to pass all of them with lots of thinking and at the end coming up with fastest and the most reliable solution. We have worked our way around difficulties while creating this roller coaster. To start the roller coaster we started with random equations within reason and connected them from a random place while following the guidelines which was also one of the difficulties we had. To face that difficult we came with many sketches and equation to put on desmos. We wanted to work in a way where we can follow the rubric guidelines and try to go above and beyond. One of the biggest difficulties we faced while creating SkyRunner was connecting the equation at a point where it doesn't look impractical and the pathway for the roller coaster makes sense mathematically. After crossing over that difficulty we didn't really have any big difficulties but when the first roller coaster was made to improve on that roller coaster was a bit hard. To improvise on something that we thought was the best was hard because to think of new ideas after looking at the already built roller coaster is hard. But at the end we came through all the difficulties we faced and here we are with our final roller coaster.

## Written Report

The first function we used is an exponential function, which is h(t)=2^(x-1.8) + 9.65. This function has a restriction on the time that states that x is less than or equal to 6.8 and less than or equal to 0 seconds, This equation starts at 10 feet.

The second function we used is a quadratic function, which is h(t)= -(x-5)(x-30) This function has a restriction on the time that states that x is less than or equal to 6.598 and 28.53 seconds, At 17.5 seconds this function reaches 156.25 feet.

The third function we used is a cosine function, which is h(t)= -9cos(0.2*pi*x)+40 This function has a restriction on time which is that x must be less than or equal to 48.8 and 28.53 seconds, This function has two cycles and has a maximum height at 49 feet when the time is at 35 seconds and 45 seconds. The minimum height of this function is 31 feet at the times of 30 and 40 seconds.

The fourth function we used is a cubic polynomial function, which is h(t)=0.1x(x -51.9)+80This function has a restriction on the time that states that x must be less than or equal to 53 and 48.8 seconds, . At 51.9 seconds, the function is at its highest point, 80 feet.

The fifth function we used is a log function, which is h(t)=-30log(x-52.5)+66.5 This function has a restriction on the time that states that x must be less than or equal to 60 seconds and it must be less than or equal to 53 seconds, The roller coaster drops from 80 feet to 40.6 feet at the end of this particular circuit in the roller coaster.

The sixth function we used is a linear function, which is h(t)=40.6 The roller coaster train approaches this function at 60 seconds at leaves at 67.3 seconds resulting in the following restriction, . The roller coasters train is 40.6 feet above the ground at this interval.

The seventh function we used is a cubic polynomial function, which is h(t)=-0.02x(x-81)^2+300 This function has a time interval that starts at 67.12 seconds and ends at 89.04 seconds which results in the following restriction, This function reaches the maximum height of 300 feet at 81 seconds.

The eighth function we used is another cosine function, h(t)=13cos x+181 This function has a restriction on the time that states that x is less than or equal to 93.31 and 89.18, . This function has a minimum height of 168 feet whilst at the time of 29π.

The ninth function we used is a cubic polynomial with the equation of h(t)= -0.1x((x-94))^2+191 This function has a restriction on the time that states the x must be less than or equal to 97.8 seconds and 93.4 seconds, This function begins at 191 feet and drops to 50 feet.

The tenth function we used is another cubic polynomial;h(t)= 3.5(-x+100)^3+10 This function has a restriction on the height that states that the height must be less than or equal to 49.58 feet and it must be less than or equal to 10 feet, . This function ends at 100 seconds.

## Calculating the Average and Instantaneous rate of change

Solve for the exact time(s) when your roller coaster reaches a height of:
• 250 feet
The roller coaster is at 250ft twice at 75.24 seconds and 86.38 seconds based on the graph.
• 12 feet
The roller coaster is at 12ft also twice at 3.03 seconds and 99.17 seconds.

Calculate average rate of change from:
• 10 to 15 seconds
• 50 to 60 seconds
Calculate instantaneous rate of change at:
• 35 seconds

## Conclusion

The roller coaster took some amount of time and hardwork to design but the outcome shows that anyone would fun on this ride.