# Rollercoaster Model

## OUR PLAN

The planning of this roller coaster model’s graph was an interesting task, with the final product completed within 2-3 days. The maximum value of x (time) as 100 seconds and y (height) as 300 feet was given, these guidelines helped to plan the organization of the graph and serve as a starting point. The intention when planning the graph was to create a graph that not only displays all the required functions but blends them together in such a manner where the graph look like a realistic roller coaster model. So, first, a rough draft of the roller coaster was created. Then, we simply graphed the functions on Desmos by creating accurate equations accordingly to the rough graph (and made certain corrections on Desmos, i.e. domain or range of certain functions). Just like any other rollercoaster, this model began with a straight line, a linear function (which was at y=10), from where the height slowly increased to the only major drop of the rollercoaster, at the maximum value of y=300. Majority of the functions were not at all difficult to incorporate into the roller coaster model with the exception of the logarithmic function; as the slope was quite difficult to restrict, while smoothly connecting to the next function. Connecting the functions in the graph smoothly was a difficult task in general, which required us to incorporate many decimal numbers to get an accurate graph. Lastly, to make the roller coaster look more appealing, a 3D effect was added by slightly altering the original equations of functions.

## General Description of Functions (of height V.S time)

Linear: y=10

• This equation is a linear function that acts as the starting point of the ride. It is at the minimum height of 10 feet (y-value), and the roller coaster moves along this line for 5 seconds (domain). This function has a y-intercept of (0,10).

Linear: y= 14.5(x-5) +10

• This is another linear function incorporated into the graph which acts as the ramp that allows the roller coaster to travel up, towards the first drop. The ramp takes the roller coaster from the bottom of the ride (10 feet) to the summit of the ride, which is at 300 feet . The travel of the ride upwards is 20 seconds long (from 5 seconds to 25 seconds on the x-axis) and the height is increased by 290 ft (10 ft to 300 ft on the y-axis).

• This quadratic equation represents the second half of the summit, at which point the roller coaster will begin to descend. The entire function is not shown as only half of the function was needed to act as the latter half of the summit. This is why there is a restriction. This part of the ride lasts 5 seconds as, just like any other roller coaster, the ride will travel very quickly as it falls. After the drop, the ride will be brought down to a height of 155 feet, at which another function is introduced. This function lasted from 25 seconds to 30 seconds on the x-axis and descended from 300 feet to 169 feet.

Polynomial: y=.1(x-30)(x-45)(x-36)(x-41)+155

• This polynomial function is connected to the preceding quadratic function to continue the descending part of the ride. This functions begins 30 seconds into the ride, at a height of 155 feet. The ride is descended to a height of 60.8 feet and then raised again to a height of 189.59 feet. Yet again, it descends to a height of 116.92 feet and rises once more to 155 feet (height at which function began). This function lasts from 30 seconds to 45 seconds on the x-axis.

Logarithmic: y= 70 log (x-44) + 155

• This logarithmic function is vertically stretched by a factor of 70 and acts as a small ramp to bring the roller coaster up to the next part of the ride. The function is shifted 155 (ft) units up and 44 units (seconds) to the right. It acts as a ramp which lasts 5 seconds (45 seconds to 50 seconds on the x-axis). At 50 seconds, the ride is half-way through, as the total time of the ride is only 100 seconds. The function begins at a height of 155 ft and increases to a height of 210 ft, from where the new function commences.

Quadratic: y= -.5(x-50) (x-60) + 209.47

• This function acts as a continuation of the logarithmic function, the small ramp that brings the roller coaster to a height of 222 feet. This function lasts from 50 seconds to 55 seconds on the x-axis (5 seconds). It is a ramp that increases the height form 210.5 ft to 222 ft (on the y-axis). Certain horizontal and vertical translations were made to the function; it was translated 209.47 units (ft) up and 50 units to the right. Not only that, but this function is reflected in the x-axis.

Linear: y = 221.97

• This linear function is a ramp, which travels from 56 seconds to 60 seconds on the x-axis (lasts 4 seconds). It sits at a steady height of 222 ft on the function.

Sinusoidal: y= - 35 cos 72 (x-60) + 256.97

• This function is a sinusoidal function, specifically a cosine function. The function represents the loop-de-loops of the ride. Loop-de-loops cannot be made on a height/time graph as it will make it seem as if the ride is going back in time, which is impossible. Therefore, these functions serve as a representation of the loops. This function lasts from 60 seconds to 75 seconds on the x-axis. The height is increased and then decreased 3 times, from 225 ft to 292 ft. Not only that, but the function is reflected in the x-axis.

Rational: y = (1/(x-81) + 222.1366667

• This function is a small ramp that the roller coaster will travel along before it descends for the end of the ride. It is shifted 222.14 units (ft) up and 81 units (seconds) to the right. It travels from 75 seconds to 80 seconds on the x-axis (lasts 5 seconds), and sits at a steady height of 222 ft on the function.

Quadratic: y = -2 (x-80)^2 + 221.14

• This quadratic starts to bring the roller coaster down from a height of 222 feet to a height of 44.4 feet. This part of the ride lasts 9 seconds, from approx 80 seconds to 90 seconds. It is translated 221 units (ft) up and 80 units (seconds) to the right. Not only that, but this function is reflected in the x-axis.

Exponential: y = 2^-(x-94.5) + 9.977902913

• This exponential function is the last 11 seconds of the ride (from 89 seconds to 100 seconds). It continues bringing the roller coaster down to its original height of 10 feet (from 45 feet).

## Calculations

Q. Solve for time when roller coaster reaches the height of

a) 250 feet

b)12 feet

Q. Calculate average rate of change from:

a) 10 to 15 seconds

b) 50 to 60 seconds

Q. Calculate instantaneous rate of change from at:

a) 35 seconds

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