ADDITIONAL MATHEMATICPROJECT WORK 2015

ACKNOWLEDGMENT First and foremost, I would like to thank my Additional Mathematics teachers, Puan Siti Rafida and Puan Haznida as they gives us important guidance and commitment during this project work. I also would like to give thanks to all my friends for helping me and always supporying me while completing this project. Without them this project would never have had its conclusion. Thanks to myself because struggling to finish up this project. For their strong support, I would like to express my gratitude to my beloved parents for supplying the equipment and money needed to complete this project. They have always been by my side and I hope they will still be there in the future.

OBJECTIVEWe students taking Additional Mathematics are required to carry out a project work while we are in Form 5. This project have to be done individually. Upon completion of the Additional Mathematics project work,we are gain valuable experience and able to:

* Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems.* Experience classroom environments which are challenging,interesting, meaningful and hence improve their thinking skills.* Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems.* Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected.* Experience classroom environments that stimulates and enhances effective learning.* Acquire effective mathematical communication through oral and writing and to use the language of mathematics to express mathematical ideas correctly and precisely.* Enhances acquisition of mathematical knowledge and skills through problem solving in ways that increase interest and confidence.* Prepare ourselves for the demand of our future undertakings and workplace.* Realize that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.* Use technology especially the ICT appropriately and effectively.* Realize the importance and the beauty of mathematics.

INTRODUCTION

In modern mathematics,a function is defined by its set of inputs, called thedomain; a set containing the set of outputs, and possibly additional elements, as members, called itscodomain; and the set of all input-output pairs, called itsgraph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called theimageof the function). For example, we could define a function using the rulef(x) =x2by saying that the domain and codomain are thereal numbers, and that the graph consists of all pairs of real numbers (x,x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are calledfunction spaces, the properties of which are studied in such mathematical disciplines asreal analysis,complex analysis, andfunctional analysis.In analogy witharithmetic, it is possible to define addition, subtraction, multiplication, and division of functions, in those cases where the output is a number. Another important operation defined on functions isfunction composition, where the output from one function becomes the input to another function.

TYPE OF FUNCTION

Pierre De Fermat

PIERRE DE FERMAT ; 17 August 1601 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to innitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of nding the greatest and the smallest ordinates of curved lines, which is analogous to that of the dierential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Fermats TheoremPIERRE DE FERMAT developed the technique of adequality (adaequalitas) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in mathematical analysis. According to Andr Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word (parisots) to refer to an approximate equality. Claude Gaspard Bachet de Mziriac translated Diophantus's Greek word into Latin as adaequalitas.[citation needed] Paul Tannery's French translation of Fermats Latin treatises on maxima and minima used the words adquation and adgaler.

Fermat used adequalityfirst tofind maxima of functions, and then adapted it to find tangent lines to curves. To find the maximum of a term p(x), Fermat equated (or more precisely adequated) p(x) and p(x+e) and after doing algebra he could cancel out a factor of e, and then discard any remaining terms involving e. To illustrate the method by Fermat's own example, consider the problem of fi nding the maximum of p(x)=bx-x^2. Fermat adequated bx-x^2 with b(x+e)-(x+e)^2=bx-x^2+be-2ex-e^2.

(a) (i) Mathematical Optimization Inmathematics,computer science,operations research,mathematical optimizationis the selection of a best element from some set of available alternatives. PART 1

(ii) Global Maximum/Minimum Global maximum is the highest point over the entire domain of a function or relation. While, global minimum is the lowest point over the entire domain of a function or relation. (iii) Local Maximum/Minimum Local maximum can also be expressed as Relative Maximum. It is a greatest value in a set of points but not highest when compared to all values in a set. The set points can be global maximum. Local minimum is the least value that locates within a set of points which may or may not be a global minimum and it is not the lowest value in the entire set. It can also be termed as Relative Minimum.

(b) i-Think Map

PART 2(a) x

yLet the length = y m Sub x = 25 into (1), breadth= x m y = 100 2 (25) = 50 mPerimeter = 200 m Dimension = 25 m x 50 m 4x + 2y = 200 Sub x = 25 and y = 50 into (2), 2x + y = 100 Amax = 25 m x 50 m y = 100 2x (1) = 1250 m Area, A = xy (2) Maximum Area = 1250 m Sub (1) into (2), A = x(100 2x) = 100x 2x = 100 4xA is maximum when = 0, 100 4x = 0 4x = 100 x = 25 m

(b) Dimension of the box = (30 2h) x (30 2h) x h Volume, V = h(30 2h) = h[900h 120h + 4h] = 900h 120h + 4h = 900 240h + 12h V is maximum when = 0, 900 240h + 12h = 0 12, h 20h + 75 = 0 (h 5) (h 15) = 0 h = 5, h = 15 If h = 15, Length = 30 2(15) = 0 Not a valid answer. h = 5 Sub into V, V = 5[30 2(5)] = 5(20) = 2000 cm Largest possible volume = 2000 cm

(i) Sketch the graph of function P(t).PART 3

x0123456789101112

y024190018002700335836003358270018009002410

(ii) When does the mall reach its peak hours and state the number of people. The mall reach peak hours after 6 hours of opening which is 3.30 pm with 3600 people at the mall.

(iii) Estimate the number of people in the mall at 7.30 pm. 900 people in the mall at 7.30 pm.

(iv) Determine the time when the number of people in the all reaches 2570. The time when the number of people in the mall reaches 2570 are 1.18 pm and 5.42 pm.

FURTHER EXPLORATION

(a) Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of infinitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists. Linear programming can be applied to various fields of study. It is used in businessandeconomics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation energy, telecommunications, and manufacturing. It hasa proved useful in modelling diverse types of problems in planning, routing, scheduling, assignment and design.

How it started?

LEONID KANTOROVICH

The problem of solving a system of linear inequalities dates back at least as far as Fourier, who in 1827 published a method for solving them and after whom the method of FourierMotzkin elimination is named. The first linear programming formulation of a problem that is equivalent to the general linear programming problem was given by Leonid Kantorovich in 1939, who also proposed a method for solving it. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. About the same time as Kantorovich, the Dutch-American economist T. C. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel prize in economics. In 1941, Frank Lauren Hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later Simplex method; Hitchcock had died in 1957 and the Nobel prize is not awarded posthumously. During 1946-1947, George B. Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force. In 1947, Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases. When Dantzig arranged meeting with John von Neumann to discuss his Simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent. Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948. Postwar, many industries found its use in their daily planning. Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the observable universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm. The theory behind linear programming drastically reduces the number of possible solutions that must be checked. The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.

(b)(i)(a) Total Cost RM1400 100x + 200y 1400 100, x + 2y 14 Total Space 7.2 0.6x + 0.8y 7.2 x 5, 3x + 4y 36 Ratio: 3x 2y 2y 3x

(b) x + 2y = 14 x 0 14

y 7 0

3x + 4y = 36 x 0 12

y 9 0

x 0 4

y 0 6

2y = 3x

(ii) Finding maximum storage volume First Method: Find all corner points Corner Points are: 1. (0,0) 3. (8,3) 2. (12,0) 4. (3.5, 5.25) We want to maximize storage volume according to the function: V = 0.8x + 1.2

Corner Points0.8x + 1.2Answer(s)

(0,0)0.8(0) + 1.2(0)0

(12,0)0.8(12) + 1.2(0)9.6

(8,3)0.8(8) + 1.2(3)10

(3.5,5.25)0.8(3.5) + 1.2(5.25)34.3

We will then subtitute the values from each corner points to find the maximum storage value.

Therefore, the maximum storage volume is 10. To get this value we have to use 8 cabinets of x and 3 cabinets of y to achieve maximum storage value.Second Method: Using set squareIn this method, we use the assistance of a set square to find the maximum storage volume. To do this, we have to use and objective function. ax + b = k

So, the value is 0.8x + 1.2 = k. we put in the value of k as 10 to get the new equation 8x +12 = 10. We then have to draw a line. The new graph is as follows;

From the new line, with the aid of a set square, we move it from the new line until it touches the last point in the feasibility region. The last point is (8,3).Therefore, the maximum storage is; k = 0.8(8) + 1.2(3) = 10Therefore, the maximum storage volume is 10. To obtain 10, we have to use 8 cabinets of x and 3 cabinets of .

Cabinet x(iii)

Total price of cabinetBalance moneyCabinet can be purchaseTotal areaTotal volumeFinal combination

4RM 400RM 1000 = 5(4 0.6) + (50.8)= 6.4(4 0.8) + (51.2)= 9.24x + 5yArea = 6.4 mVolume = 9.2 m

5RM 500RM 900(5 0.6) + (40.8)= 6.2(5 0.8) + (41.2)= 8.85x + 4yArea = 6.2 mVolume = 8.8 m

6RM 600RM 800 = 4(6 0.6) + (40.8)= 6.8(6 0.8) + (41.2)= 9.66x + 4y Area = 6.8 mVolume = 9.6 m

7RM 700RM 700(7 0.6) + (30.8)= 6.6(7 0.8) + (31.2)= 9.27x + 3yArea = 6.6 mVolume = 9.2 m

8RM 800RM 600 = 3(8 0.6) + (30.8)= 7.2(8 0.8) + (31.2)= 108x + 3yArea = 7.2 mVolume = 10 m

9RM 900RM 500(9 0.6) + (20.8)= 7(9 0.8) + (21.2)= 9.69x + 2yArea = 7 mVolume = 9.6 m

iv) Justification If I was Aaron, I would choose the fifth combination, which is 8 cabinets of x and 3 cabinets of My reasons are as follows;

- It uses the maximum space that can be held in the office room, which is 7.2 metres squared. - It can store up to 10 cubic metres of file, which is the maximum storage volume. - It comes at a reasonable price at RM 1 400, which is not too expensive. - It is the most suitable combination for the future of Aarons company.

REFLECTION

Ive found a lot of information while conducting this Additional Mathematics project. Ive learnt the uses of function in our daily life.

Apart from that, Ive learnt some moral values that can be applied in our daily life. This project has taught me to be responsible and punctual as I need to complete this project in a week. This project has also helped in building my confidence level. We should not give up easily when we cannot find the solution for the question.

Then, this project encourages students to work together and share their knowledge. This project also encourages students to gather information from the internet, improve their thinking skills and promote effective mathematical communication.

Lastly, I think this project teaches a lot of moral values, and also tests the students understanding in Additional Mathematics. Let me end this project with a poem;

In math you can learn everything, Like maybe youll like comparing, You have to know subtraction, a.k.a brother of addition, You might say I already simplified, so now your work aint jank edified, So now dont think negative, Its better to think positive, Dont stab yourself with a fork, But its better to show your work, My math grades are fat, But not as fat as my cat, Lets get typical, And use a pencil, Add Math is fun!

CONCLUSION

After doing research, answering question, drawing graph, making conjecture, conclusion and some problem solving, I realized that Additional Mathematics is very important in daily life.

But, what is the use of calculus in daily life of normal people like us? In reality most people are not going to use the function in daily life. Having a firm understanding of the funtion as with most maths helps increasing logical thinking, critical thinking and number sense.

About this project, overall is quite joyfull and interesting because I need to plan it carefully and systematic because it is about my future. In fatc, the further exploration is a good session because it can open my mind about calculus.

In a nutshell, I can apply the concet and skills that I had in solving problems in Additional Mathematics, I think this project is very beneficial for all students.