# Mother Nature

### Experience natures four elements with our roller coaster!

## What is a roller coaster?

## The Nature Elements Rollercoaster

-The sinusoidal function represents water.

-The odd degree three polynomial function represents fire.

-The exponential and quadratic function at the beginning represent land (volcano).

-The sinusoidal function towards the end represents air.

## Rough Draft

## Equations

## 1:Exponential Growth function

The functions are numbered according to the number on Desmos.

Exponential Growth - h(t)= 2^(t+3.35), {10 ≤ y ≤ 169.49}

- This function contributed to the element of land which was represented by a volcanic shape. The exponential function created one side of the volcano. The exponential function was ideal because of it's horizontal asymptote and long, upward stretch.

## 2: Quadratic function

Quadratic - h(t)= 0.734[(t-7.65)^2] +160

-This function served is also part of the land element which we characterized by the volcano. The parabola's concave shape was suitable for the crater of the volcano

## 3: Exponential function

Exponential - h(t)= 1.99^(-t+18.546)

-Like the first exponential function, this one also serves as the side of the volcano and connects to the sinusoidal function

## 4: Sinusoidal function

Sinusoidal - h(t)= -40sin[0.99(t-12.5960] + 60

- The sinusoidal function was used to replicate waves which represents the element of water. The function was ideal because of the its peaks and troughs.

## 5: Logarithmic function

Logarithmic - h(t)= 69.9log(t-39) +55.25

-The logarithmic function is used to as part of the representation of fire. It is part of the imagery of flames.

## 6: Rational function

Rational Function - h(t)= 257.68/(t-50.89)^2 +100

-This function was used to represent fire through flames. Because flames have an irregular pattern, there are various functions that make up the fire portion of the roller coaster, and generally lack a sense of symmetry.

## 7: Parabolic function

Parabolic Function- h(t)= -9.14[(t-50.9)^2] +270

-This function is part of the fire element and serves as a connection between the rational and polynomial function

## 8: Polynomial function

Polynomial** **–h(t) = -18.8 (t-51.4)^5 +260

-This function represents the fire element and connects the parabola and the rational function.

## 9: Polynomial function

-This function represents the fire element and connects the parabola and the rational function.

## 10: Quadratic function

-This function represents the fire element and connects the two polynomial functions.

## 11: Polynomial function

- This is the last function that represents the fire element before the next sinusoidal function begins.

## 12: Sinusoidal function

- This sinusoidal function represents the element of air because it portrays the fact the air flows freely.

## 13: Linear function

-This function is also part of the air element and it is to transition from the air to the ground (earth).

## 14: Exponential function

**+ 10**

-This function is the last component, the end of the roller coaster.

## Finding Essential Values (a, k, m, b)

## Method of Calculation

## Equation #2: Green Parabola

h(t)=a(x-7.65)^2+160

169.48=a(4.055-7.65)^2+160

169.48-160=a(-3.595)^2

9.48/12.924025=a

0.734=a

h(t)=0.734(x-7.65)^2+160

## Equation # 3: Green Exponential Function

h(t)=b^(-x+18.546)

169.07057=b^(7.402)

logb169.07057=7.402

log169.07057/logb=7.402

0.301008918=logb

10^0.301008918=b

1.99=b

h(t)=1.99^(-x+15.546)

## Equation # 4: Blue Sinusoidal Function

Min=20

Amplitude=max-min/2

=100-20/2

=80/2

=40

h(t)=-40(k(x-12.596))+60

Equation of axis=max+min/s

=100+20/2

=120/2

=60

Point(12.59870026, 59.892011) Period:6.347

h(t)=-40sin(k(x-12.596))+60

59.892011=-40sin(k(12.59870026-12.596))+60

59.892011-60=-40sin(0.00270026k)

-0.107989/-40=sin(0.00270026k)

0.002699725=sin(0.00270026k)

sin^-1(0.002699725)=0.00270026k

0.002699728/0.00270026=k

0.99=k

2π/period=k

2π/6.347=k

0.99=k

## Equation # 5: Orange Logarithmic Function

h(t)=alog(x-39)+55.25

88.02366=alog(41.94-39)+55.25

88.02366-55.25=alog(2.94)

32.77366=alog(2.94)

32.77366/0.46834733=a

69.9=a

h(t)=69.9log(x-39)+55.25

## Equation # 6: Red Rational Function

h(t)=a/(x-50.89)^2+100

119.3264=a/(47.3-50.89)^2+100

119.3264-100=a/(-3.66)^2

13.3956 x 19.2364=a

257.68=a

h(t)=257.68/(x-50.89)^2+100

## Equation # 7: Orange Parabola

h(t)=a(x-50.9)^2+270

269.231=a(50.61-50.9)^2+270

269.231-270=a(-0.29)^2

-0.769/0.0841=a

-9.14=a

h(t)=-9.14(x-50.9)^2+270

## Equation # 8: Orange Polynomial

h(t)=a(x-51.4)^5+260

254.5109=a(52.1814-51.4)^5+260

254.5109-260=a(0.7814)^5

-5.4891/0.291317808=a

-18.8=a

h(t)=-18.8(x-51.4)^5+260

## Equation # 9: Orange Polynomial

h(t)=a(x-56.1)^3+130

118.2383=a(54.5498-56.1)^3+130

118.2383-130=a(54.5498-56.1)^3

-11.7617/-8.9523813183=a

1.20=a

h(t)=1.20(x-56.1)^3+130

## Equation # 10: Red Parabola

h(t)=a(x-60.27)^2+300

299.516=a(60.05-60.27)^2+300

299.516-300=a(60.05-60.27)^2

299.516-300=a(-0.22)^2

-0.484/0.0484=a

-10=a

h(t)=-10(x-60.27)^2+300

## Equation # 12: Purple Sinusoidal Function

Min=100

Amplitude=max-min/2

=300-100/2

=200/2

=100

h(t)=100sin(x-64.26)+200

## Equation # 13: Purple Linear Function

Point 2(94.49887, 37.0335)

m=y2-y1/x2-x1

=37.0335-299.9742/94.49887-90.9408

=-262.9407/3.55807

=-73.9

h(t)=-73.9(x-95)

## Equation # 14: Green Exponential Function

h(t)=b^(-(x-97.5))+10

37.034-10=b^(-(94.499-97.5))

27.034=b^3.001

logb27.034=3.001

log27.034/logb=3.001

log27.034/3.001=logb

0.477144388=logb

10^0.477144388=b

3.00=b

h(t)=3^(-(x-97.5))+10

## Calculations

## Height vs. Time

__Height: 12 feet__

h(t)=2^t+3.35

log2 12=t+3.35

3.58=t+3.35

3.58-3.35=t

0.23496=t

The roller coaster is at 12 feet at 0.23 seconds.

__Function # 14: Green exponential function__

h(t)=3^(-(t-97.5)) +10

12-10=3^(-(t-97.5))

2=3^(-(t-97.5))

log3 2=-t+97.5

0.63-97.5=-t

-96.87/-1=t

96.87=t

The roller coaster is at 12 feet at 96.87 seconds.

__Height: 250 feet__

__Function # 6: Red rational function__

h(t)=257.68/(t-50.89)^2 +100

250-100=257.68/(t-50.89)^2

150=257.68/ t^2-101.68t+2584.71

(150) x (t^2-101.68t+2584.71)=257.68

t^2-101.68t+2584.71=257.68/150

t^2-101.68t+2584.71=1,72

t^2-101.68t+2584.71-1.72=0

t^2-101.68t+2584.71=0

Then apply the quadratic formula and get the value of the t.

t=49.53

t=52.15

The equation is at 250 feet at 49.53 seconds and at 52.15 seconds.

__Function # 9: Degree five function__

h(t)=1.20(t-56.1)^5+130

250=1.20(t-56.1)^5+130

250-130=1.20(t-56.1)^5

120/1.2=(t-56.1)^5

5√100=√(t-56.1)^5

2.51=t-56.1

2.51+56.1=t

58.61=t

The equation is at 250 feet at 58.61 seconds.

__Function # 11: Degree three function__

h(t)=-(t-63.3)^3+270

250=-(t_63.3)^3+270

250-270=-(t-63.3)^3

-20=-(t-63.3)^3

-20/-1=(t-63.3)^3

3√20=3√(t-63.3)^3

2.71=t-63.3

2.71+63.3=t

66=t

The roller coaster is at 250 feet at 66 seconds.

__Function # 12: Sinusoidal function__

h(t)=100sin(t-64.26)+200

250=100sin(t-64.26)+200

250-200=100sin(t-64.26)

50/100=sin(t-64.26)

0.5=sin(t-64.26)

sin^-1(0.5)=t-64.26

0.52=t-64.26

0.52+64.26=t

**64.78**=t

Period: 6.29

t=64.78+6.29

=**71.07**

t=71.07+6.29

=**77.36**

t=77.36+6.29

=**83.65**

t=83.65+6.29

=**89.94**

t=(-0.52+64.26) + 3.15

=63.74+3.15

=**66.89**

t=66.89+6.29

=**73.18**

t=73.18+6.29

=**79.47**

t=79.47+6.29

=**85.76**

The bolded answers are the values shown on the roller coaster. This function has many more time values at a height of 250 feet which includes these numbers +/- 6.29 (2**π).**

__Function # 13: Last linear function__

h(t)=-73.9(t-95)

250/-73.9=-73.9(t-95)/-73.9

-3.38+95=t

91.62=t

The equation is at 250 feet at 91.62 seconds.

## Average Rate of Change

__10 to 15 seconds__

10 < x < 15

(10, 164.1), (15, 32.4)

__164.1 – 32.4__

10 – 15

= __131.7__

-5

= -26.34

__50 to 60 seconds: __

50 < x < 60

(50,262.6), (60, 299.3)

__299.3 - 262.6__

60 - 50

=__36.7__

10

=3.67

## Instantaneous Rate of Change

IROC at 35 seconds

h(t)=-40sin(0.99(t-12.596))+60

h(35)=-40sin(0.99(35-12.596))+60

=-40sin(22.18)+60

=-40(-0.19)+60

=7.51+60

=67.51

(35, 67.51)

h(35.001)=-40sin(0.99(35.001-12.596))+60

=-40sin(0.99(22.405))+60

=-40sin(22.18095)+60

=-40(-0.1886643889)+60

=67.546555585

(35.001, 67.546555585)

IROC=y2-y1/x2-x1

=67.546555585-67.51/35.001-35

=0.036555585/0.001

=36.556

Therefore the instantaneous rate of change at 35 seconds is 36.556 feet/ second.

## Summary

Functions are not only used to create man made objects but can also be found in nature. As such, our group decided to model our roller coaster after the four core elements of the earth; land, air, fire and water appropriately naming it 'Mother Nature'. This task was simple, but there were some restrictions we had to follow which included the fact that it must be created using the seven functions we have learned about, it must stay within a 100 second time limit and must be between 10 to 300 feet in height. The creation of the roller coaster was time consuming and required a lot of thinking and analysis. We started our assignment with a rough sketch of the roller coaster which is based on the four elements of earth: land, water, fire and air. To follow this theme, we categorized the functions as to what element we felt that they would represent. For example, we easily decided on the fact that a sinusoidal function would represent water. Using this method, we created a rough design of what we would like our roller coaster to look like. Unlike water, we had some difficulties using functions to represent the other elements. For example, we had trouble on deciding what function could represent air. We first wanted to make a tornado like spiral that would symbolize air but we were not able to do so. Therefore, we had to change our ideas and we resulted in using another sinusoidal function to represent air. After we had a basic idea of how the roller coaster would look and what functions would be used, we began to create the equations in desmos. To do so, we would put in a base equation of the function we needed and added transformations to it according to where would like it to be. In one case, we put in the base equation for an exponential function; h(t) = 2^t and then we put in transformations such as h(t) = 2^t +10. We added 10 as our c value because we needed the roller coaster to have a minimum height of 10 feet. Using this idea, we continued making all the equations for each function but it wasn’t easy. This is because it was hard to connect the equations with another so that they would intersect and the roller coaster would flow as one large piece-wise function. This involved us changing some of the values in equations and creating domain and range restrictions. Creating the restrictions was also difficult for the same reason. We needed to find restriction values that would allow functions to intersect but not cause the function to be long or too short. This lead to having restriction values with many decimal places to have an exact point of intersection. Overall, we enjoyed creating this roller coaster because it gave us creative freedom and allowed us to review everything we have learned in this course.