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The speed of Ada and Fay

Extracts from this document...

Introduction

IB Student

DP1 HL Mathematics

Internal Assessment Type II

My Internal Assessment

In this internal assessment, I am going to create a recursive formula for a mathematic model. In this investigation, I use real life examples and uses real data to finish my investigation. Afterward, I place my model to real life case study. However, there are different limitations in different cases. Some of the limitations are dimensional plane, speed, boundaries and more. At last, I am going to present my recursive formula, image00.png.

The speed of Ada and Fay

After reading the Internal Assessment question sheet, I had an idea on using a similar real life case study. Then, I invite my cousin, Ada and her dog, Fay, to have a running test and let me collect a set of data for my Internal Assessment. Therefore, I set a straight track with a distance of 100 meters and they need to run for 10 times. So, I can collect a set of accurate data. After each of them run for 10 times, I will take an average time for the 100 meters run and then uses the speed formula,image01.png. In this formula, V stands for velocity, which is the speed; the S stands for distance, at last, T stands for the period of time the runner runs. Below is Ada’s data for running 100 meters:

Ada’s running records

Trails

Time (s)

1

16.6

2

16.7

3

16.8

4

16.7

5

16.6

6

16.5

7

16.6

8

16.8

9

16.5

10

16.7

After collecting Ada’s set of data, then I need to define the average time of Ada running a 100 meters track, in the other word, the mean of the data. Therefore, I am going to use the mean equation from the statistic chapter, which isimage12.png. The calculation will be as follow,

image22.png

From the calculation, I define that the average time of Ada running 100 meters is 16.7 seconds.

...read more.

Middle

There are different methods finding the distances of the lines. First, for lines “A1F2”, “A0F1” and “A1F1”, I am going to use the Python’s theorem. The calculation of the lines will shown below,

image02.png

After finding the distance of “A0F1”, then I got enough information to use the Python’s theorem to find the length of line “A1F1”.

image03.png

The length of “A1F1” is 65.115282 meters. Below will show the step to find “A1F2”.

image04.png

After calculation, I define the length of line ”A1F2” is 49.1153 meters. After finding this lines’ length, then I can use the propriety of similar triangle to define the length of “Y2F2” and “X2F2”.

image05.png

By finding “Y2F2”, we can know the Y-coordinate of Fay’s “F2” position.

image06.png

After finish calculating the length of lines, “F2X2” and “Y2F2”, now I got a full coordinate of Fay’s position in point “F2”, which is (48.2740, 2.9486).

Result of the diagram after finding all the unknowns

image07.png

For other points, it is using the same methods to determinate their coordinates. I am going to show the calculation below to define the coordinate of point “F3”. Afterward I will use the result to define a recursive formula for x- and y-coordinates of Fay’s position to establish a discrete mathematical model.

Calculation for “F3” coordinate

image08.png

Using Python’s theorem, to find “A2F2”,

image09.png

Finding “A2F3”,

image10.png

Using the propriety of similar triangle to define the length of “Y3F3”,

image11.png

By using Python’s theorem, to define the length of “a”, which “a” is equals “F3X3-F2X2”.

image13.png

Finding “F3X3”,

image14.png

After finding length of line “F3X3” and “Y3F3”, then I have calculated the coordinate of Fay’s position after six seconds, which is (33.6078, 9.3441).

...read more.

Conclusion

Conclusion

Throughout the internal assessment, I use real data instead of creating it. In this assessment, I include different methods, equation to help me to define the mean speed of Ada and Fay, to calculate the position of Fay and form a recursive formula and more.

In this assessment, I also use different kind of computer software to present my mathematic mode and finding solutions. I use “Grapher” in Mac to plot different graphs to solve problem and looked closely to the situation I got on my hand.  

image40.png

Snapshot One: Using the graphing software, Grapher by Apple

Except using “Grapher”, I also use AI (Adobe Illustrator) to draw the out line of Fay’s and Ada’s route. Then, I printed out and use for calculation. This software gave me a very accurate diagram that makes my calculation much more accurate and faster.

image42.png

Snapshot Two: Using Adobes Illustrator, to draw the diagram

At last, I use my friend while doing mathematics, which is my calculator, Casio fx-9860GII SD. The GDC help me to find the logarithm function for simulating the situation. In addition, it helps me to calculate different kinds of math problems while I am doing my internal assessment.

Due to the help of technology and my mathematic textbook, which was written by Smythe. I finally form the recursive formula to the mathematical model in this internal assessment, which is image00.png

Bibliography

  • Adobe Illustrator. Vers. CS4. United States: Adobe, 2010. Computer Software.
  • Fx-9860GII SD. Casio Computer Co/LTD. Tokyo. Japan. GDC
  • Grapher. Vers. 2.1(43). United States: Apple Inc, 2005. Computer Software.
  • Science, Design. Microsoft Equation Editor. Computer software. Vers. 3.o. Web. 20 March. 2010.
  • Smythe, Peter. Mathematics - Text and Teacher's Guide Higher and Standard Level With Higher Level Options. Grand Rapids: Mathematics, 2005. Print.

...read more.

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