IA4 - Jacky Chong IB Student D1 HL Mathematics Internal...

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Jacky Chong IB Student D1 HL Mathematics Internal Assessment Type II - Colored December 28, 2010 Internal Assessment – Modelling a Functional Building In this paper, I modeled the route traced by a pet dog named Buddy as he follows his owner Angie. With the use of Trigonometry, I successfully generated correct recursive formulae: 2 1 2 1 1 1 ] ) 1 [( - - - - + - - - = n n n n n X Y ut n X vt X X and 2 1 2 1 1 1 ] ) 1 [( ] ) 1 [( - - - - + - - - - + = n n n n n X Y ut n Y ut n vt Y Y . I proved them logically. Finally, I applied my model on a real-life situation. Contents: Part I - Introduction I.a What Is The Situation?. ............................. 2 I.b Variables and Constraints. ......................... 3 Part II – The Recursive Formulae II.a Coordinates Of Several Points Of Buddy. ............. 3 II.b Patterns And Proof. ............................... 5 Part III – Application Of My Discrete Mathematical Model III.a Simulation Of A Variety Of u, v, t, d Values. ........ 8 III.b Buddy Catches Up With Angie!. .................... 14 III.c Assumptions And Limitations Of 1
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Model. ............ 15 Part IV - Real-life Situation IV.a Conveyor System – Does It Work?. ................... 16 IV.b The New Model. ................................... 17 Part V – Conclusion. ..................................... 20 Bibliography . .......................................... 2 2 2
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Part I: Introduction In this paper, I will model the route traced by Buddy, a pet dog, as he is chasing his owner, Angie. In this paper, I will mathematically model a building similar to the one shown in Figure 1. I will then design an office block inside this building. The Running Route Of Buddy And Angie Figure 1 : A is Angie and B is Buddy. Angie runs towards the North direction along the straight seashore. At the beginning, Buddy is d meters to the East of Angie. Both runners start from rest (A 0 and B 0 ). Angie runs at a constant speed of u ms -1 . Buddy notices Angie and runs towards her at constant speed v ms - 1 . In attempt to catch Angie, Buddy runs directly towards her. He changes direction once per t second(s) when he looks up to notice Angie’s new position (as shown in 3
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figure 1). 4
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I.b Variables and Constraints There are several variables and constraints in the following investigation: The variables are the initial distance between Angie and Buddy, d ; Angie’s speed, u ; Buddy’s speed, v ; and the time Buddy changes direction once, t . The first constraint is that all variables must not be zero. If u and v are zero, Angie and Buddy are both not moving; if t is zero, Buddy is not changing direction, and he is probably not moving also (we will see why); if d is zero, Buddy has already caught Angie before they start running. All these situations are meaningless in our investigation. Other constraints will be explored later in
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This document was uploaded on 12/31/2011.

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IA4 - Jacky Chong IB Student D1 HL Mathematics Internal...

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