The Levi(m)athan

Rollercoaster Summative By: Dhrumil, Harsh, Kandarp, Marut

Rollercoaster rough sketch

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The above graph is made up of 12 equations. It has 1 rational function, 3 linear functions, 1 quadratic function, 1 cubic function, 2 sinusoidal functions, 2 quartic functions, 1 logarithmic function and 1 exponential function.

How we created Levi(m)athan

We decided to use the basis of real life rollercoasters such as Leviathan and Behemoth in order to give us an idea of what a rollercoaster should consist of. By using pictures of different rollercoasters, we created a rough sketch of our future rollercoaster. Since we knew that the rollercoasters rely on momentum in order to function, our first drop is our highest drop. We started out with base functions of each equation and adjusted the transformations based on how we wanted to transition throughout the rollercoaster itself. However the equations did not naturally intersect properly so we had to place restrictions on them. The challenge was figuring out which values allowed us to intersect at one specific point and not overextend. It was difficult to find the exact values for the restrictions since we mainly used trial and error until we were content with the solutions. While there were many difficulties with this assignment we persevered and successfully completed our assignment.

General Description of the Height vs Time of Levi(m)athan

In the height versus time graph that we created, we worked hard towards creating an aesthetically appealing graph. This ended with us exceeding the minimum amount of equations required in order to make the rollercoaster look realistic. Levi(m)athan starts off at a height of 15ft. It proceeds to rise to the maximum height of 300ft within 12.9 seconds. As it is rising it maintains a steep slope up until 285.7ft. At the instantaneous height of 300ft the coaster is neither gaining or increasing height therefore there is 0 rate of change. Upon reaching the maximum height levi(m)athan starts to rapidly accelerate and descend in height to 20ft in 3.17 seconds. It is rapidly increasing it's rate of change as well. At the instantaneous height of 20ft the coaster stops descending and starts to ascend in a sinusoidal manner. After completing two cycles of the sinusodial transformed function it stats to ascend to 159.42ft. While it is ascending the coaster outlines the cubic function in 28.245 seconds. During the rise to 159.42ft the function decreases the rate of change until 92ft. At 92ft the coaster moves forward in time but is not gaining or losing altitude. This means that the coaster has no vertical instantaneous rate of change at 92ft for 0.56 seconds. After 0.56 seconds the coaster starts to rapidly increase in the vertical rate of change up until it intersects with the quartic function at (48.246,159.57). The quartic functions ascends to an altitude of 190ft while decreasing the vertical rate of change. At 190ft the coaster again moves forward in time but is not gainig or losing altitude for 0.34 seconds. After 0.34 seconds the coaster starts to descend in altitude until it reaches 69.26ft. At 69.26ft the quartic function intersects with the sinusodial function. The coaster descents to 25ft, ascends to 105ft and descends again to 25ft in the span of 15.02 seconds where it intersects with the last quartic function. The quartic function displays similar characteristics as the first quartic function. The coaster ascends to 90ft where it once again has a vertical instantaneous rate of change of 0 for 0.30 seconds. The quartic descends to 48.98ft where it intersects with the logarithim function. The function descends to 10.00ft at a steady negative rate of change until we reach the minimum height required. At 10.00ft the logarithim function intersects with a linear function. The coaster then proceeds to reach a 100 seconds while not increasing or decreasing in altitude.


EQUATIONS:

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Rational Function

y = -70 (1/x - 9) + 5


  • Reflected in the x-axis
  • Vertically Stretched by a factor of 70
  • Shifted right by 9 units
  • Shifted up 5 units
  • Domain: {XER l 2 <= x <= 7.6}

Linear Functions

y = 15


  • There is no slope, and the linear function has been shifted by up 15 units
  • Straight Line
  • Domain: {XER l 0 <= x <= 2}


y = 10


  • There is no slope, and the linear function has been shifted by up 10 units
  • Straight Line
  • Domain: {XER l 92.7 <= x <= 100}


y = 73x - 500


  • There is a slope of 73 and the linear has been shifted down by 500 units
  • Diagonal line (helps the coaster to climb the mountainous climb to 300ft)
  • Domain: {XER l 7.6 <= x <= 10.8}

Quadratic Function

y = -2.8 (x - 13)^2 + 300


  • Reflected in the x-axis
  • Vertically stretched by a factor of 2.8
  • Shifted right by 13 units
  • Shifted up 300 units
  • Domain: {XER l 10.8 <= x <= 13}

Exponential Function

y = -7^(x - 12.2) + 305


  • Reflected in the x-axis
  • Vertically Stretched by a factor of 7
  • Shifted right by 12.2 units
  • Shifted up 305 units
  • Domain: {XER l 13 <= x <= 15.05}

Sinusoidal Functions

y = 100 cos (4/5(x - 20)) + 120


  • Vertically Stretched by a factor of 100
  • Amplitude: 100
  • Horizontally stretched by a factor of 5/4
  • Shifted by 20 units to the right
  • Shifted up by 120 units
  • Equation of Axis: 120
  • Domain: {XER l 15.05 <= x <= 32.54}



y = 40 sin (3/5(x + 13)) + 65


  • Vertically Stretched by a factor of 40
  • Amplitude: 40
  • Horizontally stretched by a factor of 5/3
  • Shifted by 13 units to the left
  • Shifted up by 65 units
  • Equation of Axis: 120
  • Domain: {XER l 54.87<= x <=69.84}

Polynomial Functions

y = 1/8 (x - 40.1)^3 + 92 - Cubic Function (Degree 3)



  • Vertically compressed by a factor of 1/8
  • Shifted to the right by 3 units
  • Shifted up 92 units
  • Domain: {XER l 32.54 <= x <= 48.24}



y = - 1/1.89 (x - 51)^4 + 190 - Quartic Function (Degree 4)



  • Reflected into the x axis
  • Vertically compressed by a factor of 1/1.89
  • Shifted right by 51 units
  • Shifted up by 190 units
  • Domain: {XER l 48.24 <= x <= 54.87}



y = -0.9 (x - 72.6)^4 + 90 - Quartic Function (Degree 4)



  • Reflected into the x axis
  • Vertically compressed by a factor of 0.9
  • Shifted right by 72.6 units
  • Shifted up by 90 units
  • Domain: {XER l 69.94 <= x <= 75.14}

Logarithmic Function

y = - 20 log (x - 75) + 35



  • Reflected into the x axis
  • Vertically stretched by a factor of 20
  • Shifted right by 75 units
  • Shifted up by 35 units
  • Domain: {XER l 75.2 <= x <= 92.7}

Calculations

Solve for 12 Feet

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Therefore, at 89.125 seconds the roller coaster is at a height of 12 feet.

Solving for 250 Feet

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Therefore, at 10.274 seconds and 14.259 seconds the roller coaster is at a height of 250 feet.

Average Rate of Change

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Therefore, between the interval of 10 to 15 seconds the average rate of change of the roller coaster is -31.484 feet/seconds .

Instantaneous Rate of Change

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Therefore at the the instant of 35 seconds, the rate of change is 4.7 feet/seconds.

Conclusion

Much like this roller coaster, MHF4U had its ups and down but in the end the ride was incredible thanks to the greatest math teacher ever: Mrs. Danielle Brignull - Tabone.