Mathematical Project Term 3 Name: Abdelrahman Abubakir Rashid Mohammed ali Ahmed Abdullah Mohammed Class:12/03

Roller Coaster

TASK 1: Equation Modeling 1. The SLIDE. This ride is located in the kiddie section! The height of the rider above ground, h yards, after t seconds can be modeled by the function: h(t)= a.How long does the ride last (from starting height to reaching ground level)? h(t)=0 0=-0.5t+40→0.5t=40→t=40/(0.5) =80 S

b. change the numbers so that the ride starts higher and drops faster. h(t)-4t+200 c. Now how long does the ride last, based on the changes in part b? h(t)=0 0=-4t+200→4t=200→t=200/4 =50

Question 2: The LITTLE DROP On this ride, for some period of time, the rider dips below the ground level. The height of the rider after t seconds can be modeled by the function: h(t)=4t 2 -44t+96

a. What is the starting height of this ride? t=0 so h(0)=4(0)^2-44(0)+96=96 h(0)=96

b. How long is the rider below the ground? h(0)<0 will be between the roots h(t)=4t^2-44t+96=0 (t-8)(t-3)=0→t=8 and 3 h(t)<0 when t is between 3 and 8 so he is 5 second below the ground c. If the ride lasts a total of 10 seconds, what is the height of the exit gate? h10=4(10)^2-44(10)+96 =4(100)-44(10)+96 = =56 m

d. Use the information from the parts above to sketch a graph of the height of the ride over time with the appropriate labels on the axes. h(t)=8t-44

Question 3: The SCREAM This ride lasts for 8 seconds. The height of the rider can be modeled by the function: h(t)=(-6t 2 +12t)-(t 2 -12t+32)

a.At which height does this ride begin? t=0 h(t) = (-6t 2 +12t)-(t 2 -12t+32) h(0) = (-6(0) 2 +12(0)) - ((0) 2 -12(0)+32)-(0)t 4 = 0m b. At what height does this ride end? T=8 H(8) = (-6(8) 2 +12(8)) - ((8) 2 -12(8)+32) =0m

c. At what height is the ride after 5 seconds? h(5)=(-6(5)^2+12(5))((5)^2-12(5)+32) h(5)=270 meter d. At what time(s) does the ride hit ground level? h(t)=0 / t=? 0 = (-6t 2 +12t)-(t 2 -12t+32) -6t 2 +12t = 0 or t 2 -12t+32 = 0 T=2 sec, t=0 sec t=8 sec, t=4 sec The ride hits the ground at t=0,2,4,8 seconds.

Sketch a graph of the ride with the appropriate labels on the axes. You may use you’re the Grapher program and experiment with the WINDOW to get the right picture.(copy paste your graph below)

f. Using the graph from part e, over what interval(s) of time does the ride drop below ground level. From 2 seconds to 4 seconds

Task 2: The Design A roller coaster can be based on mathematical functions, but they are more likely to be made up of pieces, each of which is a different mathematical function. This allows much more flexibility. Use piecewise functions to design your own roller coaster. (You will actually be modeling the height of the roller coaster over time.) Include a graph of the roller coaster. Your roller coaster MUST fit the following criteria:

A.Each of the pieces are connected to each other B.Uses at least 5 DIFFERENT functions/pieces H(t)= 0 0≤x<3 x-3 3≤x<6 -x+9 6≤x<9 0.5x-4.5 9≤x<12 1/4x ≤x<15

Once you have decided on a design of your roller coaster and have graphed it answer the following questions using the graph and the function. A.What is its starting height? 0 b. How long does the ride last? 15 c. What is its ending height? 2.25

d. Approximate how long it takes to reach its highest point? 3 e. Why would it be useful to be able to have equations for a roller coaster? so, the roller coaster should be continues line.

f. What kinds of things might you be able to figure out about the roller coaster? We can calculate the acceleration, Velocity, Slope and gravity those things should be calculated to know that Roller coaster will not be pulled by the gravity force and it will run in smooth way. g. How might it help you to design or change the design? As I have mentioned the help to design a smooth and sharp way to the roller coaster, the design should have as mush safety as possible to the rider. The design can be changed by changing the formulas making it wider, higher and sharper.