RollerCoaster
Fast and the Furious
Fast and the Furious
Summary: Creation Of RollerCoaster
The rollercoaster was created through the use of Desmo’s graphing technology and through the use of several different functions. The rollercoaster began with a rough draft on a piece of graph paper with a maximum of 300 feet and a minimum of 10 feet, as well as a time restriction at 100 seconds. The draft consisted of a miniature sketch of the rollercoaster, outlining the desired shape and design. A list of basic equations were then transformed in order to create the shape. First, on Desmos, we listed each of the parent functions that were required to complete the rollercoaster. Then, we began transforming each equation in order to follow the layout of our original sketch and design. Sometimes equations had to be changed in order to fit within the time frame of 100 seconds. We also had to add new equations to ensure that the rollercoaster flowed and followed a basic order. The most important aspect was adding restrictions and making sure that the different equations touched and did not overlap. Adding the restrictions was the hardest part of the process and it became very time consuming in order to perfect the rollercoaster. However, the finish product turned out as planned and the overall process worked well.
Written Report: First Equation
The first equation that initiates the rollercoaster is a cubic function. The a-value is a positive value of 1/6.5, which means that it is a vertical compression because (a<1). There is no horizontal shift to the graph, as the rollercoaster begins at zero seconds. Also, there is a vertical shift 10 units upwards, to ensure the rollercoaster is above ground. There is a maximum point at 25.3 and minimum at 10. The degree of the function is three, meaning it is an odd degree function and has point symmetry. The end behaviours of a degree three function includes as x approaches positive infinity, y approaches positive infinity. And when x approaches negative infinity, y approaches negative infinity. The domain of the function is {XER, X≥0} and the range of the function is {YER, y<25.3}. X≥0 is a time restriction and Y≤25.3 is a height restriction. The function is supposed to extend from quadrant 3 to quadrant 1 because it is a degree three function and it has a positive leading coefficient however the restrictions prevent this from showing on the graph.
Written Report: Second Equation
The second equation that follows up the rollercoaster is a sine function. The sine function has a positive amplitude of 120. The maximum value is 260 and the minimum value is 60. The function is shifted to the left 177.5 units and the equation of axis {vertical shift) is 140 units up. The k-value of the function is 1/5, meaning there is a horizontal stretch by a factor of 5. The period (2π/k), is 31.42. The time restriction, on the x-axis, consists of x being more than 4.63 seconds but less than 34.6 seconds. Meaning the domain of the function is {XER, 34.6≥X≥4.63}. The range of the function is {YER}. We used the sine graph for the first incline and decline because it has an appearance of a wave.
Written Report: Third Equation
Written Report: Fourth Equation
Written Report: Fifth Equation
Written Report: Sixth Equation
Written Report: Seventh Equation
Written Report: Eighth Equation
Written Report: Ninth Equation
Written Report: Tenth Equation
Written Report: Eleventh Equation
Written Report: Twelve Equation
Written Report: Thirteenth Equation
Calculations
Solve for the exact time(s) when your rollercoaster reaches a height of: 250 ft
Solve
y=250sin(1/5x+35.5)+140
250=120sin(1/5x+35.5)+140
250-140=120sin(1/5x+35.5)+140
110/120=sin(1/5x+35.5)
0.916=sin(1/5x+35.5)
sin–1(0.916)=1/5x+35.5
1.157=1/5x+35.5
Solve for Quadrant 1
1.157-35.5=1/5x
-34.343=1/5x
-34.343*5=1x
-171.715/1=x
x=-171.715
Period of y=120sin(1/5x+35.5)+140
=2π/k
=2π/1/5
=31.415
Add period
-171.715+31.415+31.145+31.415+31.415+31.415+31.415
=16.79
Solve for Quadrant 2
π-1.157=1.984
1.984=1/5x+35.5
1.984-35.5=1/5x
-33.516=1/5x
-33.516*5=1x
-167.58/1=x
x=167.58
Period of y=120sin(1/5x+35.5)+140
=2π/k
=2π/1/5
=31.415
Add period
-167.58+31.415+31.145+31.415+31.415+31.415+31.415
=20.91
Solve for the exact time(s) when your rollercoaster reaches a height of: 250 ft
Solve
y=-7.7(x-46.3)^2+300
250=-7.7(x-46.3)^2+300
250-300=-7.7(x-46.3)^2
-50=-7.7(x-46.3)^2
-50/-7.7=(x-46.3)^2
6.49=(x-46.3)^2
√(6.49)=x-46.3
+/-(2.547)=x-46.3
Solve with positive value of 2.547
+2.547=x-46.3
+2.547+46.3=x
x=48.85
Solve with negative value of 2.547
-2.547=x-46.3
-2.547+46.3=x
x=43.75
Solve for the exact time(s) when your rollercoaster reaches a height of: 12 ft
Solve
y=2.25(x-70.80)^2+10
12=2.25(x-70.80)^2+10
12-10=2.25(x-70.80)^2
2/2.25=(x-70.80)^2
0.88=(x-70.80)^2
√(0.88)=x-70.80
+/-(0.938)=x-70.80
Solve with positive value of 2.547
+0.938=x-70.80
+0.938+70.80=x
x=71.74
Solve with negative value of 2.547
-0.938=x-70.80
-0.938+70.80=x
x=69.86
Solve for the exact time(s) when your rollercoaster reaches a height of: 12 ft
Solve
y=1/6.5(x)^3+10
12=1/6.5(x)^3+10
12-10=1/6.5(x)^3
2=1/6.5(x)^3
2*6.5=1(x)^3
13=1(x)^3
3√(13)=x
x=2.351
Solve for the exact time(s) when your rollercoaster reaches a height of: 12 ft
Solve
y=-1/80(x-100)^3+10
12=-1/80(x-100)^3+10
12-10=-1/80(x-100)^3
2=-1/80(x-100)^3
2/-1/80=(x-100)^3
2*-80/1=(x-100)^3
-160=(x-100)^3
3√(-160)=x-100
-5.43=x-100
-5.43+100=x
x=94.57
Calculate average rate of change from: 10-15 seconds
y=120sin(1/5x+35.5)+140
y=120sin(1/5(10)+35.5)+140
y+120sin(37.5)+140
y=116.26
y=120sin(1/5x+35.5)+140
y=120sin(1/5(15)+35.5)+140
y=120sin(38.5)+140
y=226.16
y2-y1/x2-x1
=226.16-116.26/15-10
=109.9/5
=21.98
Calculate average rate of change from: 50-60 seconds
50 to 60 seconds
y=360/(x-47.48)-10y=360/(50-47.48)-10
y=360/(7.9365)-10
y=142.857-10
y=132.86
y=60cos(1/2x-24.6)+95
y=60cos(1/2(60)-24.6)+95
y=60cos(5.4)+95
y=133.08
AROC= y2-y1/x2-x1
=133.08-132.86/60-50
=0.22/10
=0.022
Calculate instantaneous rate of change at: 35 seconds
y=1/4(35-34)^3+19.95
y=1/4(1)+19.95
y=20.2
35.001 seconds
y=1/4(x-34)^3+19.95
y=1/4(35.001-34)^3+19.95
y=1/4(1.001)^3+19.95
y=1/4(1.003)+19.95
y=20.20075
y2-y1/x2-x1
=20.20075-20.2/35.001-35
=0.00075/0.001
=0.75