ADDITIONAL MATHEMATICS

PROJECT YEAR 2011

FORM5

NAME : ONG KAH HONGFORM : 5 CEMERLANGNO I/C : 940422-06-5075TITLE :Crude OilTEACHER : MISS PHUA CHUI CHUI

CONTENTS1. Introduction

Introduction of project…………….

Introduction of integration………...Definition of integration…………..History of integration……………..

2. Problem SolvingPart1……………………………….Part2……………………………….Part3……………………………….Part4……………………………….Part5……………………………….

Further Exploration………………....3. Conclusion……………………..

4. Reflection……………………....

INTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK1/2011

The aims of carrying out this project work are to enable students to :a)Apply mathematics to everyday situations and appreciate theimportance and the beauty of mathematics in everyday lives b)Improve problem-solving skills, thinking skills , reasoning andmathematical communication c)Develop positive attitude and personalities and instrinsicmathematical values such as accuracy , confidence andsystematic reasoning d)Stimulate learning environment that enhances effectivelearning inquiry-base and teamwork e)Develop mathematical knowledge in a way which increase students’ interest and confidence.Introduction of integrationIn mathematics,integration is a technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign “∫,” as in

∫f(x), usually called the indefinite integral of the function. (The symbol dx is usually added, which merely identifies x as the variable.) The definite integral, written

with a and b called the limits of integration, is equal to g(b) − g(a), whereDg(x) = f(x).Some antiderivatives can be calculated by merely recalling which function has a given derivative, but the techniques of integration mostly involve classifying the functions according to which types of manipulations will change the function into a form the antiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/(x + 1) can be easily recognized as the derivative of loge(x + 1). The antiderivative of (x2 + x + 1)/(x + 1) cannot be so easily recognized, but if written as x(x + 1)/(x + 1) + 1/(x + 1) = x + 1/(x + 1), it then can be recognized as the derivative of x2/2 + loge(x + 1). One useful aid for integration is the theorem known as integration by parts. In symbols, the rule is ∫fDg = fg − ∫gDf. That is, if a function is the product of two other functions, f and one that can be recognized as the derivative of some function g, then the original problem can be solved if one can integrate the product gDf. For example, if f = x, and Dg = cos x, then ∫x·cos x = x·sin x − ∫sin x = x·sin x − cos x + C. Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

Definition

The process of finding a function, given its derivative, is called anti-

differentiation (or integration). If F'(x) = f(x), we say F(x) is an anti-derivative of f(x).

Examples

F(x) =cos x is an anti-derivative of sin x, and ex is an anti-derivative of ex.

Note that if F(x) is an anti-derivative of f(x) then F(x) + c, where c is a constant (called the constant of integration) is also an anti-derivative of F(x), as the derivative of a constant function is 0. In fact they are the only anti-derivatives of F(x).

We write  f(x) dx = F(x) + c.

if F'(x) = f(x) . We call this the indefinite integral of f(x) .

Thus in order to find the indefinite integral of a function, you need to be familiar with the techniques of differentiation.

HISTORY

Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was

remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers.

Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy.

Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music.

Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lebesgue (1875-1941) put definite integration on a firm logical foundation.

Liouville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm.

In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms. Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev.

In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made in extending his method to certain classes of special functions.

The capability for definite integration gained substantial power in Mathematica, first released in 1988. Comprehensiveness and accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its integration code. Besides being able to replicate most of the results from well-known collections of integrals (and to find scores of mistakes and typographical errors in them), Mathematica makes it possible to calculate countless new integrals not included in any published handbook.

Part 1Route 1.1 2.1 1.2 2.2 1.3 2.3

Distance 13km 24km √109km √307km √85km √104kmBearing Goes to

northGoes to east

N73.3 º N27.9ºW N77.5ºE N78.7ºE

Coordinates (oo)❑→

( 013) (00)❑

→

(240 ) ( 0

13)❑→

(1016) (24

0 )❑→

(1517) (15

21)❑→

(2423) (15

21)❑→

(2523)

Possible Dangers

Coral reef Shark,Infestedwater

Coral,reef,sunken ship

Shark,infested water,sunken ship,thunderstorm

Giant octopus

Giant octopus,thunderstorm

Time For route 1(1.1,1.2,1.3)= 55minutes 59secondsFor route 2(2.1,2.2,2.3)=1hour31minutes36seconds

Judging from the possible dangers & possibilities of intruding into the preserved and conservation aresas and the time taken to reach the offshore oil rig,route 1 is the recommended option

Part 2a)Starting position (o0)→(8

6)

Vresultant=Vboat+Vcurrent

(45v

35v )=(36 cos a

36 sina )+(−150 )

Vresultant=(VcosΘv sinΘ )

=(v ( 810 )

v ( 610 ))

=(45v

35v )

Vcurrent=(−150 )

35v=36sin a

V=60sin a _________①45v=36cos a-15______②

From ①, ②,we get a=22.4º,v=22.855km/h

Time taken=[( 1022.855 ) x1]hour

=0.4375hourb)From( 8

6 )→

Vresultant

Vboat

Vcurrent

- - - - - - - - - - - a

Vresultant=(VcosΘv sinΘ )

=(v2

√104

v10

√104)

Vboat=(36 cosB36 sinB )

Vcurrent=(−150 )

Vresultant=Vboat+Vcurrent

(v2

√104

v10

√104)=(36 cosB

36 sinB )+(−150 )

By using the similar concept as shown in step ①,B=54.6 º,v=29.915km/hTime taken=( √104

29.915x 1)hour=20.3416

c)From(1518)→(24

24)

Vresultant

Vboat

Vcurrent

- - - - - - - - - - - B

Vresultant=Vboat+VcurrentVcurrent=(−15

0 )Vboat=(36 cosC

36 sinC )Vresultant=(VcosΘv sinΘ )

=(v ( 9√117 )

v ( 6√117 ))

Similary,by working it out youself,C=20.3 º,v=22.548km/h

Time taken=[( √11722.548 ) x1]hour

=0.48hourd)Time to reach the wind –

farm=10.00a.m+26minutes15seconds+20minutes28seconds =10:46:43+2hours+28minutes48seconds =13:15:31a.m

Vresultant

Vboat

Vcurrent

- - - - - - - - - - - C

Part 3a) P=cAu2

C= P

Au2

=10000

10(13)2

=1000169

=5.917 b)(1)E=∫

0

t

Pdt

50000000=∫0

t

10000dt

=[ 10000 t ] t0

=10000t t=5000seconds (2)500000000=∫

0

t

cA (0.02 t )2dt

=∫0

t1000169

(10 )¿¿

=∫0

t4

169t 2dt

=[ 4169 ( t3

3 )] t0 = 4

507t 3

t=1850.6secondsPart 4 a)v=πR2h dvdt=dv

duxdhdt

Vfull=1000000000

= πR2h πR2

(3000)=100000000 πR2=100000

3

v= πR2h =100000

3h

dvdh=100000

3 __________①3000metres=(10x365x24)hoursdhdt= 3000

10x 365 x24

= 5146

dvdt=( 100000

3 )( 5146 )

=250000219 barrels per hour

b)

V=πr2h =π(0.25)2h =0.0625πh

0.25cm

dvdh

=0.0625π_______①

Vfull=π(0.25)2(1) =0.0625πTfull=(5x60)seconds

dvdt

=0.0623π5 x60

=π

4800________②

dvdt

= dvdh

xdhdt

π4800

= 0.0625πxdhdt

dhdt =

1

300ms−1

¿ 1x 100cm

(300 x1

60 )min =20cms-1

Part 5Oil Reserves - Top 20 Nations (% of Global)

Saudi Arabia has 261,700,000,000 barrels (bbl) of oil, fully 25% of the world's oil. The United States has 22,450,000,000 bbl.

The United States government recently declared Alberta's oil sands to be 'proven oil reserves.' Consequently, the U.S. upgraded its global oil estimates for Canada from five billions to 175 billion barrels. Only Saudi Arabia has more oil. The U.S. ambassador to Canada has said the United States needs this energy supply and has called for a more streamlined regulatory process to encourage investment and facilitate development.- CBC Television - the nature of things - when is enough enough

Oil Production & Consumption, Top 20 Nations by Production (% of Global)

Here are the top 20 nations sorted by production, and their production and consumption figures. Saudi Arabia produces the most at 8,711,000.00 bbl per day, and the United States consumes the most at 19,650,000.00 bbl per day, a full 25% of the world's oil consumption.

Exports & Imports

Here's export and imports for all the nations listed in the CIA World Factbook, sorted alphabetically as having exports and imports.

Conspicuously missing is the United States, but I can tell you that we consume 19,650,000.00 bbl per day, and produce 8,054,000.00, leaving a discrepancy of 11,596,000.00  bbl per day.

This compares to the European Union, which produces 3,244,000.00 bbl per day and consumes 14,480,000.00 bbl per day for a discrepancy of 11,236,000.00 per day. Basically, about the same.

World Oil Market and Oil Price Chronologies: 1970 - 2003

Further Exploration Petroleum engineers work in the technical profession that involves extracting oil in increasinglydifficult situations as the world's oil fields are found and depleted. Petroleum engineers searchthe world for reservoirs containing oil or natural gas. Once these resources are discovered, petroleum engineers work with geologists and other specialists to understand the geologicformation and properties of the rock containing the reservoir,

determine the drilling methods to be used, and monitor drilling and production operations.Low-end Salary:$58,600/yr 

Median Salary:$108,910/yr 

High-end Salary:$150,310/yr EDUCATION:

Engineers typically enter the occupation with a bachelor¶s degree in mathematics or anengineering specialty, but some basic research positions may require a graduate degree. Mostengineering programs involve a concentration of study in an engineering specialty, along withcourses in both mathematics and the physical and life sciences. Engineers offering their servicesdirectly to the public must be licensed. Continuing education to keep current with rapidlychanging technology is important for engineers.MATH REQUIRED:College AlgebraGeometryTrigonometryCalculus I and IILinear AlgebraDifferential EquationsStatisticsWHEN MATH IS USED:Improvements in mathematical computer modeling, materials and the application of statistics, probability analysis, and new technologies like horizontal drilling and enhanced oil recovery,have drastically improved the toolbox of the petroleum engineer in recent decades.POTENTIAL EMPLOYERS:About 37 percent of engineering jobs are found in manufacturing industries and another 28 percent in professional, scientific, and technical services, primarily in architectural, engineering,and related services. Many engineers also work in the construction, telecommunications, andwholesale trade industries. Some engineers also work for Federal, State, and local governmentsin highway and public works departments. Ultimately, the type of engineer determines the typeof potential employer.FACTS:Engineering diplomas accounted for 12 of the 15 top-paying majors, with petroleum engineeringearning the highest average starting salary of $83,121.

ConclusionI have done many researches throughout the internet anddiscussing with a friend who have

helped me a lot in completing this project. Through the completion of this project, I havelearned many skills and techniques. This project really helps me to understand more about the uses of progressions in our daily life.This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found.Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from theinternet, improve thinking skills and promote effective mathematical communication.Last but not least, I proposed this project should be continue because it brings a lot of moral values to the student and also test the students understanding in Additional Mathematics.

ReflectionAfter spending countless hours,day andnight to finish this Additional

Mathematics Project,here is what I gotto say:

Doing this project makes me realize howimportant Additional Mathematicsis.Also, completing this project makesme realize how fun it is and likable isAdditional Mathematics