# Mother Nature

## What is a roller coaster?

A roller coaster is a amusement park attraction which has a railroad track with many different turns and heights and is usually fast paced.

## The Nature Elements Rollercoaster

Our roller coaster is named "Mother Nature" since the idea behind the roller coaster is to experience a ride through nature's four elements.

-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.

## 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.

-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

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

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

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

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

## 11: Polynomial function

Polynomial– h(t) = -(t-63.3)^3 + 270

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

## 12: Sinusoidal function

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

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

## 13: Linear function

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

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

## 14: Exponential function

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

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

## Method of Calculation

To get the equations of the functions, we first started off with the exponential function and put restrictions on it. To get the following functions, we used the base graph of each and shifted the horizontally and vertically to a point where they could intersect the function before it which gave us the 'd' and 'c' values. To calculate the 'a' values, or the base of the exponential graphs, we chose a point that we wanted the function to hit and substituted the values into the equation to solve for 'a.' The point that we substituted was primarily where the previous function left off and this compressed/stretched the graph accordingly.

## Equation #2: Green Parabola

Point (4.055, 169.48)

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

Point(11.144, 0.7057)

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

Max=100

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

Point(41.94, 88.02366)

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

Point(47.23, 119.2364)

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

Point(50.61, 269.231)

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

Point(52.1814, 254.5109)

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

Point(54.5498, 119.2383)

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

Point(60.05, 299.516)

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

Max=300

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 1(90.9408, 299.9742)

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

Point(94.499, 37.034)

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

## Height vs. Time

Height: 12 feet

Function # 1: Green exponential function

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 (2π).

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

Average rate of change=y2-y1/x2-x1

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.