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11 Mathematical Methods Problem Solving and Modelling Task Unit 1: Algebra, Statistics and Functions Topic 2: Functions and Graphs Kyle Gabrielle Silagan

Table of Contents Introduction.......................................................................................................................................................................... 2 Observations:.................................................................................................................................................................... 2 Assumptions:....................................................................................................................................................................2 Variables:..........................................................................................................................................................................2 Independent and Dependent Variables:...........................................................................................................................3 Method................................................................................................................................................................................. 3 Developing a Solution...........................................................................................................................................................3 Gathering the Data:..........................................................................................................................................................3 Developing a Quadratic Model from the Data Collected:................................................................................................4 Refined Model:.................................................................................................................................................................6 Evaluation.............................................................................................................................................................................7 Verification of Results:......................................................................................................................................................7 Reasonableness of Results:...............................................................................................................................................7 Strengths and Limitations:...............................................................................................................................................7 Conclusion............................................................................................................................................................................8 Recommendations:...........................................................................................................................................................8 Conclusion:........................................................................................................................................................................8 Appendix...............................................................................................................................................................................9 Bibliography..........................................................................................................................................................................9

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Introduction In mathematics a function is an expression or rule which defines the relationship between the independent and dependent variable. Polynomial functions for instance, consists of one or more terms with only positive integers as exponents of variables. This can be modelled in various forms: Polynomial Function Linear Quadratic Cubic Quartic

Formula

f f f f

Degree (exponent) 1 2

( x )=mx +c ( x )=ax2 +bx +c ( x )=ax3 + bx 2 + cx +d ( x )=ax 4 +bx 3+ cx 2+ dx+ e

3 4

Quadratic equations are a second-degree polynomial function which illustrates a constant slope known as a parabola. Its general rule is depicted as:

f ( x ) =ax2 +bx +c where a, b and c are real numbers and

a≠0 .

and when factorised appears as:

f ( x ) =( x−h )2+k Where h and k are the vertex points which is also known as the turning point formula. The purpose of this report is to determine whether polynomial functions can be used to establish the optimal flight time of a whirlybird and eventually make a conclusion upon the most suitable dimension for the selected variable. A unique function must also be constructed that models how the time of flight is affected by the altered variable. When one variable is being experimented, all other variables must remain constant. Lastly, a report is to be produced about how the model was developed, modified and refined.

Observations: In this task, it can be observed that:  There are four main variables in a whirlybird and one must be modified.  A stopwatch is be used to time the optimal flight time of the whirlybird  A polynomial function must be created to model the parabolic path of the whirlybird as well its flight duration.  In order to generate a quadratic model, an appropriate graphing software must be used.  The model established must then be refined to a polynomial of any degree

Assumptions: In reference to the report conditions, it is assumed that:  The whirlybird is able to be in flight for a short period of time.  The original whirlybird template given with the task sheet will not be utilised instead, a similar one will be (See appendix A).  The whirlybird’s dimensions will remain fixed throughout the experimentation apart from the variable being changed.  The capability of the whirlybird to orbit and its stability will eventually deteriorate.  The height of its dropping point remains unchanged every time.  The angle it is released from stays as similar as possible.  The variable being changed will alter the data to form the parabola

Variables: A whirlybird is a flying model built out of folded paper which normally follows a parabolic path. However, its flying duration can be affected by multiple variables such as: Controlled variables in this task include:  The length of the wings (l)  The width of the wings (w)  The length of the whirlybird’s body (b)  The width of the whirlybird’s body (m)  The height the whirlybird is released from  The angle the whirlybird is released from.

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Uncontrolled variables in this task include:  The time the whirlybird stays in flight.  Unpredicted weather occurrence  Outside interference  The movement of the whirlybird

Independent and Dependent Variables: The independent variable is the length of the wings as it is the factor being manipulated. With multiple whirlybirds with various wing spans constructed to experiment with. The dependent variable will be the flight time of the whirlybird as this will be measured based on the modified wing length (independent variable).

Method With the aim of finding the best whirlybird dimension to longer sustain flight, each whirlybird will be modified eight times from its original dimension, having nine in total to experiment with. Each whirlybird’s wings (l) will be reduced 8 times from its original measurement by 1cm consistently (See Appendix B), whilst the remaining variables as well as its dropping point will remain unchanged. The data will be collected through timing the whirlybird’s flight span using a stopwatch as soon as it is released from 2 metres above the ground and until it reaches the floor (See appendix C). Technological mathematical procedures will also be utilised to investigate the gathered results, synthesising a quadratic model based on the features of the data collected and refining this to consider a polynomial. This involves problem solving and mathematical modelling of polynomial functions and graphs. This is through algebraic solutions which will include determining the values of each term in the general: f ( x ) =ax2 + bx + c and turning point:

f ( x )= ( x +h )2 +k

quadratic function based on the data collected as well as through substitution. Followed by graphical solutions which will be displaying a visual representation of the parabolic path the whirlybird follows, through the use of the graphical software – Desmos and Microsoft Excel.

Developing a Solution Gathering the Data: The domain of the data displays the set of all possible Length of the wings (cm) Average time (sec) values of the independent variable scaling from the 12 2.09 highest to the lowest length of the wings. The range 11 2.4 on the other hand, is the difference between the 10 2.78 highest and the lowest flight time average. Just by 9 2.94 looking at the data gathered, it is evident that the 8 3.07 whirlybird with the wing span of 8cm was the longest 7 2.89 to remain in flight with an average of 3.07 sec, 6 2.8 therefore indicates the best dimension for the chosen 5 2.47 variable (length of the wings l). 4 2.04 Figure 1: Average time the whirlybird remains in flight with various wing lengths (See appendix C) Domain = 4 < cm < 12 Range = 2.04 < t < 3.07

Obj e ct21

Figure 2: Graph displaying the flight time of the whirlybird in comparison with its wing length The graph above illustrates the parabolic path which was the result of the several experimentations upon the effect of different whirlybird wing spans to its optimal flight time. The best whirlybird dimension to remain the longest in flight will be considered as the turning point for this graph (8, 3.07).

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Developing a Quadratic Model from the Data Collected: Based on the gathered results, a polynomial function can be created which displays a parabola in correlation to the plotted points from figure 2. It is evident that the impact of the constantly modified whirlybird wing length to its flight duration illustrates a concave parabolic shape. Therefore, the turning point of the plotted data will be utilised and substituted in the quadratic turning point equation.

Creating a Basic Turning Point Formula

Calculating “a”

Turning Point = (8, 3.07)

2 f ( x )=a(x−8) +3.07

Substitute T.P to the turning point formula:

Random Point = (4, 2.04)

f ( x )=a(x−b)2 +c

Substitute random point into the turning point formula:

f ( x )=a(x−8)2 +3.07

2.04=a(4 −8 )2+ 3.07 2

2.04=a(−4) +3.07 Now that the values in the turning point formula are identified: 2

f ( x )=− 0.064(x−8)2 +3.07 This will then be used to determine the general quadratic function.

Converting the Basic Turning Point Formula into its General Quadratic Form

2.04 −2.04 =a (−4 ) +3.07 −2.04

a(−4)2=1.03 −16 a =1.03

a= f ( x )=−0.064 ( x−8 ) 2 +3.07

f ( x ) =−0.064 ( x−8 ) ( x−8 ) +3.07

1.03 −16

a=−0.064

2 f ( x ) =− 0.064(x −8 x−8 x +64 )+ 3.07 2 f ( x )=− 0.064(x −16 x +64 )+3.07

f ( x ) =−0.064 x2 +1.024 x−4.096+3.07 f ( x ) =− 0.064 x2 +1.024 x−1.026

Both general and turning point functions are now determined hence modelling the parabolic path and optimal flight time of the whirlybird.

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The general quadratic equation has now been set: intercept can be solved.

f ( x )=− 0.064 x2 +1.024 x−1.026 . Through this formula, the x and y

Use the Quadratics formula to solve ‘x’

0= y 2 0=−0.064 x +1.024 x−1.026

x=

−b ± √ b −4 ac 2a

x=

−1.024 ± √(1.024) −4 (−0.064 )(−1.026 ) 2(−0.064 )

x=

−1.024 ± √1.049 −0.26 −0.128

x=

−1.024 ± √0.789 −0.128

2

Solving the y intercept

0=x 2 0=−0.064 (0 ) +1.024 (0)−1.026

y=0+ 0−1.026 y=−1.026 2

x=

x=

Y intercept = (0, −¿ 1.026)

Turning Point Validation

x=

T.P.

−1.024 ± 0.888 −0.128

T.P.

−1.024 + 0.888 −1.024 −0.888 x= −0.128 −0.128

T.P.

x=

X intercept = (1.063, 0) and (14.97, 0)

−1.024 2 ( −0.064 )

x=

x=1.063 x=14.94

−b 2a

−1.024 −0.128 x=8

T.P.

Substitute into the general quadratic equation: 2

y=−0.064 (8 ) +1.024 (8)−1.026 y=− 0.064(64 )+ 8.192−1.026 y=−4.096+8.192−1.026 y=3.07 Turning Point = (8, 3.07)

As displayed above, through the values of the quadratic function, the x and y intercepts were solved to ensure that the graph which will be established will have the same value hence indicating that the calculations were accurate. This was also done with the turning point to confirm its validity.

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2

y=−0.064 x +1.024 x−1.026 2

R =0.9867

Figure 5: Parabola graphed through DESMOS displaying the quadratic polynomial trendline The model represents a very refined correlation between the plotted points and the quadratic polynomial linear of regression (R 2 value). The trendline follows the approximate parabolic path of each collected data points and has minimal error bars. However, considering the mathematical solutions established further up, the x intercepts from this graph (1.074, 0) (14.926, 0) does not exactly match the calculated x intercept value (1.063, 0) (14.97, 0). Whilst the y intercept and turning point precisely does. Referring back to the aim of this assignment, above is the visual representation of the optimal flight time of the whirlybird and its parabolic path. Through determining its R 2 value the estimated values for its function were shown and compared with the calculated function solved through substitution.

Refined Model: To create the most accurate model for the collected data, the degree of the polynomial must be increased further to form a parabolic path with virtually no error bars.

Quartic Regression Model Average Flight time (s)

3.5 3

f(x) = 0 x⁴ − 0.02 x³ + 0.12 x² + 0.12 x + 0.52 R² = 0.99

2.5 2 1.5 1 0.5 0

3

4

5

6

7

8

9

10

11

12

13

Length of the Wings (l) Figure 6: Graph illustrating a quartic polynomial trendline created in Microsoft Excel The function for the refined model displayed above is y = 0.0006x 4 – 0.01693 + 0.1233x2 + 0.1231x + 0.5212, featuring a fourth-degree polynomial equation, formed after its general function y = ax4 + bx3 + cx2 + dx + e. It is clear that the trendline virtually intersects each point in its parabolic path, demonstrating an outstanding correlation coefficient possible for the data collected. The R 2 value is also very close to 1 therefore, indicates the strong and accurate relationship between each point.

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Evaluation Verification of Results: The aim of this task was to establish a conclusion upon the best whirlybird dimension for the selected variable. This was being tested against the flight time of the model. Based on the data collected, it is evident that the whirlybird with 8cm in length wings was the longest to remain in flight and therefore is marked as the most suitable dimension for that variable. The solution developed to solve the problem is valid as mathematical reasoning has been utilised to display how the optimal flight duration increases up to a certain point, at which it gradually then begins to deteriorate. That point is the domain of the data which indicates the maximum time the model could fly from the range of times received. Although despite all the substitution and several calculations to ensure that the data is authentic, minimal errors have occurred whilst gathering the data and thus may have impacted a possibly longer optimal flight time which slightly reduces the accuracy of the solutions presented.

Reasonableness of Results: Based on the information presented, the results can be considered as reasonable. The mathematical reasoning applied in the developing a solution section above, together with two models of different polynomial (quadratic and quartic) trendlines, established a balanced spread of data. It compares the distinct regression value results opposing each other. With the quartic R 2 value closer to one in contrasts with the quadratic function. The gathered results were accumulated through taking the multiple variables, observations as well as assumptions into consideration prior the experimentation to ensure that the most accurate set is produced. Furthermore, the domain and range are considered as, only a certain amount of data is reasonable. Due to the limited amount of data being closed to the trendline, this indicates that all outlier data will have a low correlation hence when the centre of the graph (Vertex) is considered the data can be defined as reasonable. However, the results may also be slightly affected by uncontrolled variables such as, interference from foreign objects and the movement of the whirlybird. Additionally, it was assumed that the whirlybird flight mechanism will deteriorate over time due multiple trials causing the paper being damaged as it constantly hits the ground. This was proved correct and consequently limited the potential flight time of the whirlybird. Despite these minor inaccuracies, overall the variables and assumptions were withheld and therefore it can be argued that the results are reasonable.

Strengths and Limitations: A brief evaluation of the validity of results involving mathematical arguments on the strength and limitations of the model produced:

Strengths  Variables were predetermined, allowing the data collection to be done, with consideration on how to get the most accurate data possible.  Models were created with mathematical reasoning and digital graphical technology, which compared and refined both versions of data.  The data collected was able to form an appropriate parabola without excessive inconsistencies.  Data displayed patterns and trends as expected with parabolic graphs establishing a reasonable final result.  The uncontrolled variables were controlled as much as possible to avoid inconsistencies.

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Limitations  The area where the data collection was conducted has objects across the space which may have interrupted the flight and, caused constant retesting or damage to the whirlybird.  When solving the x intercept, the values were not precisely the same as the one displayed in the graph indicating slight inaccuracy with the calculations.  The whirlybirds flight mechanism deteriorated over time causing it to fly for less time on each test conducted.  Calculations could only be conducted to validate the first model as the second model was digitally created (Quartic polynomial through Microsoft Excel)

Conclusion Recommendations: Based on the analysis of limitations, improvements can be proposed to improve the reasonableness of results. These recommendations include:  Experimenting with more variables such as modifying the width of the wings and body, as this will truly determine the best dimension for a whirlybird.  Ensure that the environment whi...