SELANGOR ADDMATH PROJECT WORK 2015

ADDITIONAL MATHEMATICS PROJECT WORK 2015
KERJA PROJEK MATEMATIK TAMBAHAN 2015

SELANGOR

Tajuk : Differentiation : Topik 9 : Form 4
Tajuk : Trigonometry : Topik 5 : Form 5
Tajuk : Linear Programming : Topik 11 : Form 5
Skills Area : Kalkulus, Trigonometri

Formula : \[y=x^{n}\] \[\frac{\text{d}x}{\text{d}y}=nx^{n-1}\]

Pelajar diminta menyediakan kerja projek mengikut format yang diberikan.
Berikut adalah idea & cadangan untuk content task yang diberikan :

PART 1

a) Description
i) Mathematical Optimization
ii) global maximum / minimum
iii) local maximum / minumum
-Untuk menjawab soalan ini kita diminta mengulas definisi dengan bantuan ilustrasi dan gambarajah

b) Various Method find maximum and minimum value of quadratic function [i-Think mind map]
-Kita diminta untuk membina sebuah peta minda yang bertajuk Kaedah-Kaedah Mencari Maksima dan Minima bagi Persamaan Kuadratik.

PART 2

a) Dimension of the rectangle that maximise area of pen the its maximum area.
-Untuk menjawab soalan ini kita akan menggunakan kaedah cuba jaya (trial and error) untuk mendapatkan dimensi terbaik dan sesuai. Kita akan jadualkan data dan memplot graf kemudian kita akan menggunakan formula pembezaan untuk mendapatkan dimensi pagar bagi ladang biri-biri tersebut.

b) Largest possible volume of the box
-Untuk menjawab soalan ini, kita juga perlu mengunakan kaedah yang sama (trial and error) dengan jadualkan data (nilai-nilai yang berkemungkinan untuk ketinggian kotak) untuk mendapatkan estimation (anggaran) isipadu terbesar.

Anda disarankan untuk mencuba dengan menggunakan kertas

PART 3 (Shopping Mall)

Shooping Time

i) Lakar graf untuk fungsi P(t)
ii) Bilakah waktu puncak dan nyatakan bilangan pengunjung ke pusat beli belah
iii) Anggaran jumlah pengunjung pada jam 7.30 petang
iv) Nyatakan waktu apabila jumlah pengunjung mencapai 2570 orang

[Updates]
Jawapan :
i) Sila rujuk video youtube 
ii) Waktu puncak adalah : 3.30 petang dimana jumlah populasi pengunjung adalah : 3600 orang
iii) Pada jam 7.30 petang, dianggarkan jumlah pengunjung seramai 900 orang
iv) Jumlah pengunjung mencapai 2570 orang pada....
-Pada jam 1.20 petang pengunjung meningkat ke 2570 orang
-Pada jam 5.40 petang pengunjung berkurangan ke 2570 orang 

Peringatan
* Bagi 2 jawapan utk lebih selamat.. Ramai pelajar terlepas pandang.. bila kita shift cos (inverse cos).. kita akan dapat 2 jawapan. Kalau tak sure bole tengok graf.. graf pon akan bagi 2 jawapan..
* Untuk skor bahagian 3 (part 3) nie.. kalau boleh tunjuk kan seberapa cara yang banyak yang boleh. Disini cikgu dapati ada 2 cara yang boleh anda buat 

PERTAMA
Menggunakan rumus dgn menggunakan equation P(t) = - 1800 cos (π/2) + 1800 ..
[Kaedah dan cara ditunjukkan oleh ienasyeriena]



 KEDUA 
Menggunakan graf. Daripada situ anda tarik dan gariskan mengikut kehendak soalan.
Garisan yang dibuat mestilah putus-putus.. ye...
Rujuk video joshua rick utk details

Further Exploration (Office with Cabinet)

Susun atur kabinet dalam pejabat

a) Kajian mengenai Linear Programming (LP)
-Sejarah latarbelakang LP
-Kepentingan LP terhadap situasi kehidupan seharian
-2 contoh LP yang diaplikasi dalam pelbagai bidang

Suggested Answer
History Of Linear Programming

The 1940s was a time of innovation and reformation of how products were made, both to make things more efficient and to make a better-quality product. The second world war was going on at the time and the army needed a way to plan expenditures and returns in order to reduce costs and increase losses for the enemy. George B. Dantzig is the founder of the simplex method of linear programming, but it was kept secret and was not published until 1947 since it was being used as a war-time strategy. But once it was released, many industries also found the method to be highly valuable. Another person who played a key role in the development of linear programming is John von Neumann, who developed the theory of the duality and Leonid Kantorovich, a Russian mathematician who used similar techniques in economics before Dantzig and won the Nobel prize in 1975 in economics.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs emphasizes the praticality of linear programming. The computing power required to test all possible combinations to select the best assignment is quite large. However, it takes only a moment to find the optimum solution by modeling problem as a linear program and applying the simplex algorithm. The theory behind linear programming is to drastically reduce the number of possible optimal solutions that must be checked.

In the years from the time when it was first proposed in 1947 by Dantzig, linear programming and its many forms have come into wide use worldwide. LP has become popular in academic circles, for decision scientists (operations researchers and management scientists), as well as numerical analysts, mathematicians, and economists who have written hundreds of books and many more papers on the subject. Though it is so common now, it was unknown to the public prior to 1947. Actually, several researchers developed the idea in the past. Fourier in 1823 and the well-known Belgian mathematician de la Vallée Poussin in 1911 each wrote a paper describing today's linear programming methods, but it never made its way into mainstream use. A paper by Hitchcock in 1941 on a transportation problem was also overlooked until the late 1940s and early 1950s. It seems the reason linear programming failed to catch on in the past was lack of interest in optimizing.

Linear programming can be viewed as part of a great revolutionary development which has given mankind the ability to state general goals and to lay out a path of detailed decisions to take in order to 'best' achieve its goals when faced with practical situations of great complexity. Our tools for doing this are ways to formulate real-world problems in detailed mathematical terms (models), techniques for solving the models (algorithms), and engines for executing the steps of algorithms (computers and software).

Example Application of Linear Programming in Real-Life

Airlines

The airline industry uses linear programming to optimize profits and minimize expenses in their business.  Initially, airlines charged the same price for any seat on the aircraft.  In order to make money, they decided to charge different fares for different seats and promoted different prices depending on how early you bought your ticket.  This required some linear programming.  Airlines needed to consider how many people would be willing to pay a higher price for a ticket if they were able to book their flight at the last minute and have substantial flexibility in their schedule and flight times.  The airline also needed to know how many people would only purchase a low price ticket, without an in-flight meal.  Through linear programming, airlines were able to find the optimal breakdown of how many tickets to sell at which price, including various prices in between.


Airlines also need to consider plane routes, pilot schedules, direct and in-direct flights, and layovers.  There are certain standards that require pilots to sleep for so many hours and to have so many days rest before flying.  Airlines want to maximize the amount of time that their pilots are in the air, as well.  Pilots have certain specializations, as not all pilots are able to fly the same planes, so this also becomes a factor.  The most controllable factor an airline has is its pilot’s salary, so it is important that airlines use their optimization teams to keep this expense as low as possible.  Because all of these constraints must be considered when making economic decisions about the airline, linear programming becomes a crucial job.
b) Shipping company that dealing with filling cabinet with respectively sqm space and price.
i)  With given information
      a) 3 inequalities
      b) Plot and shade the graph

ii) 2 different method, find maximum storage volume.
iii) Tabulate all combination of cabinet
iv) If you were Aaron which combination will i choose
-The problem is, i am not him.

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[UPDATES]

Video Answer

Part 1a

Suggested Answer local Maximum and Local Minimum
Part 1b

Part 2a

Part 2b

Part 3

Part 4