# Design a Roller Coaster!

Rough Model:

## The Creation Process

Creating our coaster was relatively easy. Still, we did encounter many difficulties along the way. First, including 7 different functions in a certain Height vs. Time frame is easier said than done! With the help of the Advanced Functions course content, creating the functions themselves was the easy part. However, making them connect perfectly was difficult. It was increasingly frustrating connecting one function to the other, as we constantly shifted and stretched/compressed certain functions so that we could make them fit together. Placing restrictions on the time was also difficult as we had to take the maximum and minimum heights into consideration, as well as the time frame. Fortunately, we were able to perfectly connect each graph, leaving no holes (no matter how far you zoom into the points of intersection). Furthermore, achieving perfectly curvy parts in our coaster was one of our other main difficulties. Since we do not know calculus, we could not have perfect curves in our graph, but we still managed to make the final model much more realistic than our rough model. Overall, we designed a roller coaster that met the criteria of having a maximum height at 250 ft, a minimum height at 10 ft and a total length of exactly 100 seconds while using 9 functions.

## Function #1 - Exponential

The first function we decided to create is an exponential function. This function works perfectly for the beginning of a roller coaster as it shows that the roller coaster is increasing in steepness while the cart is accelerating. The restriction on time is {0<x<2} because the sudden change in steepness only lasts for a short period of time.

This photo illustrates the change in steepness of a roller coaster, similar to our graph.

## Function #2 - Linear

The second function we used was a linear function. The function is perfect for this point of the roller coaster as it portrays the constant motion upwards that the cart of roller coaster is experiencing. The roller coaster is increasing in height at a constant rate while moving in a constant motion. The restriction on time at this part of the roller coaster is {2<x<20}. This point of this roller coaster lasts 18 seconds because for the roller coaster to run for a full 100 seconds, the cart needs to achieve an increased height so that there is more speed and velocity when the cart goes down the first 'drop'.

This photo illustrates the cart of a roller coaster moving up at a constant rate, similar to our graph.

## Function #3 - Linear

The next function we used was another linear function. This function displays the roller coaster slowing down and stopping at the height of the first drop. Once the cart of the roller coaster reaches a height of 122.5 ft, it slows down and stops momentarily before it changes it's direction and goes in for the first drop. This process takes about 2 seconds which is why the restriction on time is {20<x<2}.

This image illustrates the cart of the roller coaster at the maximum height and staying at that point for a few second before the first 'drop'.

## Function #4 - Quadratic (Parabola)

The fourth function in our graph is a parabola. This represents the roller coaster going down the first drop and back up again. We choose this function because it shows how the roller coaster drops down and builds a high speed, and uses the momentum of that speed to go back up again within a short period of time. The restriction on time is {22<x<40} because although the roller coaster gains a lot of speed as it goes downhill, it eventually slows down a bit while going back up. The sharpness of the first drop was inspired by the initial drop of the 'Gravity Max' Roller Coaster, seen below.

Best, scariest and wackiest Roller coaster in Taiwan - 90 degree drop. The Gravity Max!

## Function #5 - Sinusoidal

The fifth function we chose to use was a sinusoidal function. It illustrates how our roller coaster begins with a slight stop during the uphill experience, but the roller coaster does continue going uphill this time, higher than the first hill. It eventually drops back down again, ending off with a small curve to show the abrupt change in speed. The restriction on time is {40<x<60}, one of the longer parts to the roller coaster since it has to travel to the maximum height of 300 feet.
This image is similar to the way our roller coaster achieves and leaves it's maximum height

## Function #6 - Rational

We chose to use a rational function right after the sinusoidal function. The rational function portrays the roller coaster about to increase in speed while approaching another drop. The restriction on time is {60<x<75} because this part of the roller coaster illustrates the abrupt stop that it goes through which slows it down.

## Function #7 - Polynomial

The seventh function we used was a polynomial function. It shows another drastic drop in height and high speed the roller coaster experiences. After the drop, the roller coaster slows down when it reaches a height of 30m, then it slowly goes up and back down the small hill. The restriction on time for this function is {75<x<90} because after the increased amount of speed after the drop, the coaster slows down and then builds a small amount of height before going down the smaller hill.

## Function #8 - Logarithmic

The eighth function is a logarithmic function. It conveys the roller coaster moving slowly as the end of the ride is near. The restriction on time is {90<x<95} as this point is moving at a very slow rate.

## Function #9 - Linear

The function we chose to end off with is a linear function. This function shows the roller coaster decreasing in height at a constant rate and slowing down at a constant speed. This lasts for 5 seconds, so the restriction on time is {95<x<100}.

## Solve for the exact time(s) when your rollercoaster reaches a height of 250 ft:

Therefore, the roller coaster reaches the height of 250 ft at 53.56 seconds and 46.44 seconds.

## Solve for the exact time(s) when your rollercoaster reaches a height of 12 ft:

Therefore, the roller coaster reaches the height of 12 ft at 0.45 seconds and 99.4 seconds.

## Calculate average rate of change from 10 to 15 seconds

Therefore the average rate of change from 10 to 15 seconds is
100/18 ft/second.

## Calculate average rate of change from 50 to 60 seconds

Therefore, the average rate of change from 50 to 60 seconds is -7.75 ft/second.

## Calculate instantaneous rate of change at 35 seconds

Therefore, the instantaneous rate of change at 35 seconds is 9.877 ft/sec