# 8 Kartoos

## Views From The 905

Our engineering team has decided to launch “8 Kartoos” this upcoming summer. The engineers and designers have worked collaboratively to design this thrilling but daring ride. We began this project by determining the maximum, and minimum height of the rollercoaster, and the total time of duration. We made ourselves familiar with the mathematical equations we are going to utilize to help bring our vision to actuality. There are 7 equations that we’ve decided to use to create our rollercoaster. These range from linear, quadratic, polynomial, degree 3 and above, rational, sinusoidal, exponential and logarithmic. Taking into consideration safety hazards, we were limited to making our maximum 300 and the total time duration 100 seconds (1:40) to avoid any conflict with the existing rides.. The engineers used standard equations to come up with the design of the rollercoaster, and applied transformations to match the design of the rollercoaster’s blueprint. We were influenced to keep our rollercoaster simple, but appealing to our desired riders. 8 Kartoos is a must ride that will definitely change the world of rollercoasters.

The most difficulties us engineers had for this project involved creating and transforming based on the restrictions given. We took standard equations and used them as a base or a guide. In regards to the background knowledge that we had, we were able use the process of trial and error on the desmos software to resemble and actual structure that actually looks like a rollercoaster.

## Linear Function

To begin with, our rollercoaster starts off with a linear function. The rollercoaster launches initially 10 feet above the ground which is where we got the "b" value and y-intercept of the equation, therefore b vertically shifts 10 units up. Using rise over run, the m value was taken by finding two different "x" and "y" values and subbing it into the formula of m=y2-y1/x2-x1. This equation from our rollercoaster is shortest in time as it only lasts one second, hence gives riders the thrill and an interesting start to 8 Kartoos!

## Logarithm Function

The second part of the rollercoaster is a logarithm function which only lasts 5 seconds of the ride. The logarithmic function is the function where b is any number such that "b" cannot equal one, b > 0, and x > 0. The maximum height it reaches is 76.5 feet starting from minimum height of only 50 feet. The significant transformations are vertical stretch by a factor of 34. There is also a vertical shift 50 units up which is the equation meets its surrounding equations to form the coaster.

## Exponential Function

As the rollercoaster begins to rise to the top, it is represented by a exponential function. This part of the ride only lasts 5 seconds (from the 6 to 11 second mark) with a maximum height of 173.3 feet and minimum height of 76.5 feet. The transformations it creates is a vertical stretch by a factor of 25, horizontal shift 9 units right and vertical translation 73.34 units up.

The quadratic function is the most frantic and exciting part of the roller coaster. The standardized form of a quadratic equation is ax2 + bx + c = 0, where "a" cannot equal 0. This part of the coaster is when the ride reaches the highest maximum point of 300 feet which is represented by the "c" value of the equation. The maximum point is reached 20 seconds into the ride. Additionally, there is a reflection in the x-axis which allows for the train to have the exciting drop! There is also a vertical stretch by a factor of 25/16 and an horizontal phase shift 20 units to the right since the equation is (x-20)^2.

## Polynomial Function

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of "x". The function used to create our roller coaster is a degree 3 function meaning it is a cubic. Our equation is a cubic polynomial which means it is a polynomial of degree 3. A cubic polynomial has the form f(x)=a3x^3 + a2x^3 + a1x +a0. In our equation, it has a reflection in the x-axis and a vertical translation 2.1219 units up. This is also the lengthiest part of our roller coaster. The riders experience our polynomial function for a total of 47 seconds between the timing 28 to 75 seconds.

## Sinusoidal Function

The sixth part of the rollercoaster is a sinusoidal function, which ranges from the 75-second to the 91-second mark, making it last for a total of 16 seconds. The sine equation has a maximum height of 253.3 feet and has a minimum height of 153.2 feet. It has been vertically translated 203.2 units up and vertically stretched by a factor of 50. The negative at the front of the equation lets us know that there is a reflection in the x-axis. It has been horizontally stretched by a factor of (2π/15). The period of this function was obtained through the formula 2π/K, which equals to 15.

## Rational Function

To end off the thrilling ride, a rational function is used, which is the last 9 seconds of the whole rollercoaster, reaching a maximum height of 182.9 feet and ending at the minimum height of 10 feet. Since x cannot equal to 90, there is a vertical asymptote there. The transformations are applied by a stretch of 192.1 units away from origin, horizontal shift 90 units to the right and vertical shift 9.21 units down.