ADDITIONAL MATHEMATICS PROJECT WORK 2011

Oil RigsMathematics in Petroleum Revealed

Content

Introduction 4Project Findings 5

Part I 6Part II 8Part III 12Part IV 13Part V 14Further Exploration 23

Conclusion 24

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“A special gift for my beloved teacher,Ummi Sharifah Fatzhiyah Syed Ahmad.

My supportive parent and friends”

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IntroductionIn realizing the nation’s very vision, the 2020 Vision, it is very demanding for Malaysia to develop the economy and technology as well as improving the competency of public infrastructure to fulfill the needs of citizens of developing country. It is no doubt that this costs lots of energy. Malaysia, a newly developed country, also one of the members of G15 countries is fortunate to possess vast reserve of energy, thanks to the vast oil reserve in Kelantan, Terengganu, Sabah and Sarawak.

The works in petroleum industry always roughly is viewed to be some kind of very tough tasks, done in very high risks. Despite the everyday physical constraints faced by workers in this field, there are more problems to be considered. These include finding the fastest way to get to the source of fuel, determining the depth of oil reservoir, and calculating the amount of energy generated by the fuel found.

One would not have it in mind that these jobs have a lot to do with Mathematics, which involves the usage of differential and integral calculus, geometry and trigonometry. In this project work, we will reveal the importance of mathematics in committing these tasks, starting from the boat’s departure to the oil rigs to the usage of calculus in finishing the operations at the oil rig.

In Part V, we will observe the oil production in major producers of petroleum worldwide. This section will also reveal the significance of petroleum usage as one of the most important source of energy.

Lastly, in Further Exploration, we shall learn something about career opportunity in petroleum industry and how a strong base in Additional Mathematics would help.

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Project Findings

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Part I

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O

5

10

15

20

x

y

25

5 10 15 20 25

Suggested route = √82+62 + √22+102 + √92+62

= 31.0147 km

Alternative route I = √102+162 + √152+172

= 42.2927 km

Alternative route II =√82+62 + √22+102 + √42+92

= 30.0469

Suggested Route Alternative I Alternative IIDistance (km) 31.047 42.2987 30.0469

Bearings053.13°011.31°056.31°

032.01°056.31°

053.13°071.57°066.04°

Coordinates to change course

(0,0)(3,6)

(10,16)(15,18)

(0,0)(10,16)(15,18)

(0,0)(8,6)

(10,16)(15,19)

Possible dangers and possibilities of intruding into the

preserved and conservation areas

May drift to coral reefs and sunken ships, also may be attacked by giant

octopus

May intrude into preservation and

conservation areas, coral reefs and may be attacked by giant

octopus

May drift to coral reefs and sunken ships, also may

encounter stormy weather

Time estimation

Total travelling time

=31.04735

=53.17 min

Total time taken= 2 hours 53.13 min

Total travelling time

=42.298735

=72.51 min

Total time taken= 3 hours 12.51 min

Total travelling time

=30.046935

= 51.51 min

Total time taken= 2 hours 51.51 min

The Alternative Route II is the best route because it has the shortest distance and fewer dangers compared to suggested route and Alternative Route I. The suggested route has the same possible dangers with Alternative Route II but it has longer time estimation which is 1.62 minutes.

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15 km/h

36 km/h

N

Part IISpeed of any moving devices, preferably motorized vehicles can roughly be determined by taking the reading of speedometer. In reality, the reading shown is actually based on the engine thrust, not how much the vehicle had travelled per unit time. The speed shown could be the actual speed of the vehicle only if no unbalanced force acting on it.

In this part, the boat is about to depart from the starting point to wind farm. After two hours, the boat will continue its course to the oil rig.

For the first part of the boat’s motion, the speed of the boat travelling from (0, 0) to (8, 6) appears to be resultant velocity as a result of combination of the speed of wind of 15

Let the resultant velocity = v1

α 1=90 °−53.13 °

¿36.87 °

β1=180 °−36.87 °

¿36.87 °

Using Cosine Rule

362=v12+152−2 ( v1) (15 )cos143.13 °

v12−2 (15 ) ( v1 )cos143.13 °+152−362=0

v1=−[−2 (15 )cos143.13 ° ]±√[−2 (15 )cos 143.13° ]2−4 (1 )(152−362)

2(1)

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15 km/h

36 km/h

N

v1=22.86∨−46.86

But v1>0

Therefore, v1=22.86km /h

Hence, the time taken to travel along the first path, t 1 is

t 1=1022.86

¿26.25min

Time taken for the boat to travel from (8, 6) to (10, 16) and from (15, 18) to (24, 24) can also be calculated by similar manner, that is

From (8, 6) to (10, 16)

α 2=90 °−11.31°

¿78.69 °

β2=180 °−78.69 °

¿36.87 °

Using Cosine Rule

362=v22+152+2 (v2 ) (15 ) cos101.31 °

v22−2 (15 ) ( v2 )cos101.31°+152−362=0

v2=−[−2 (15 )cos101.31 ° ]±√[−2 (15 )cos101.31 ° ]2−4 (1 )(152−362)

2 (1)

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15 km/h

36 km/h

N

v2=29.92∨−35.8

But v3>0

Therefore, v3=22.54km /h

Hence, the time taken to travel along the second path, t 2 is

t 2=10.229.92

¿20.45min

From (15, 18) to (24, 24)

α 3=90 °−56.31 °

¿33.69 °

β3=180 °−33.69 °

¿36.87 °

Using Cosine Rule

362=v32+152+2 (v3 ) (15 ) cos146.31 °

v32−2 (15 ) ( v3 )cos146.31 °+152−362=0

v3=−[−2 (15 )cos 146.31° ]±√[−2 (15 )cos146.31° ]2−4 (1 )(152−362)

2(1)

v3=22.54∨−47.51

But v3>0

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Therefore, v3=22.54km /h

Hence, the time taken to travel along the third path, t 3 is

t 3=10.8722.54

¿28.93min

With t 1 and t 2it is expected that the boat will reach the wind farm at

10.00+¿

After two hours of committing maintenance at the wind farm, the boat will approximately reach the oil rig at

12.47+28.93min¿1316hours

Hence, the team will make it to attend the 2.00 p.m. meeting at the oil rig, at least 44 minutes earlier.

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Part III

(a)

Given that

P=cA u2

Rearranging the equation to make the machine constant, c as the subject, we get

c= P

A u2

Consequently, we get

c= 10000

10×132

¿5.9172

(b)

We know that energy generated, E

E=∫0

t

P dt where P=cA u2

(i)

When u=13 , P=10000

50000000=10000t

t=5000 s

(ii)

Whenu (t )=0.02 t, P=5.917 (10 ) (0.02 t )2=0.02367 t 2

50000000=∫0

t

0.02367 t2dt

¿0.07889 t 3=50000000

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t=¿1850.57s

Part IV

(a)

From the information given, 100 000 000 barrels of oil are to be produced in 10 years time. This gives the rate of oil production to be

10000000010×365×24

=1191.55barrel of oils/hour

(b)

The volume of the cylinder, V is

V=252×100π

¿62500π

Hence, the rate of change of volume is

dVdt

=62500 π5

¿12500π /min

From the relationship between volume of cylinder, V and height of the cylinder, h, we can obtain the first derivative relating V and h, that is

V=π r2h

dVdh

=2πr

¿2(25)π

¿625 π

From one of the differential calculus rule, the Chain rule, the rate of increase of oil level, dhdt

can be obtained by

dVdt

=dVdh

×dhdt

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dhdt

=12500 π625 π

¿20cm /min

Part VPetroleum Worldwide

Petroleum, in present is a very significant source of energy. Apart from Malaysia, other there are vast spread of this fossil fuel all over the globe. Despite the fact that Native Americans were the first to discover the significances of petroleum, the reserve of oil is mainly spread at the countries of Middle East. For this part, we will concentrate more on the oil reserves of the United States of America, Russia and Saudi Arabia.

United States Petroleum

Petroleum in the United States has been a major industry since shortly after the oil discovery at Oil Creek, Pennsylvania in 1859. As of 2008, the US was the world's third-largest oil producer (after Saudi Arabia and Russia), producing 8.5 million barrels of oil and natural gas liquids per day.[1] The leading oil-producing area in the US in 2008 was the federal zone of the Gulf of Mexico, which produced 1.15 million barrels per day, followed by Texas (1.09 million barrels per day), Alaska (0.68 million barrels per day) and California (0.59 million barrels per day).

In 2008, petroleum was the largest source of energy in the US, providing 38% of the energy consumed (an additional 24% was from natural gas). In 2008 the US consumed 19.5 million barrels per day of petroleum products, of which 46% was gasoline, 20% diesel fuel and heating oil, and 10% liquified petroleum gas. The US imports 57% of the petroleum it uses. The largest sources of US imported oil are (in descending order): Canada, Saudi Arabia, Mexico, and Venezuela.

More than 1.8 million people work in the US petroleum industry, which includes the production, processing, transportation, and marketing of natural gas and petroleum products.

Production

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Consumption

Import

United States doesn’t export their petroleum..

Export

# Importing Nation (2006) (103bbl/day) (103m3/day)

1 United States 1 12,220 1,943

2 Japan 5,097 810

3 China 2 3,438 547

4 Germany 2,483 395

5 South Korea 2,150 342 Oil Reserves

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Russia Petroleum

The petroleum industry in Russia is one of the largest in the world. Russia has the largest reserves, and is the largest exporter, of natural gas. It has the second largest coal reserves, the eighth largest oil reserves, and is the largest exporter of oil. It is the third largest energy user.

Russia is the largest oil producer in the world, producing an average of 9.93 million barrels of oil per day in 2009 for a total of 494.2 million tons. It produces 12% of the world's oil and has a same share in global oil exports. In June 2006, Russian crude oil and condensate production reached to the post-Soviet maximum of 9.7 million barrels per day (b/d). Exceeding production in 2000 by 3.2 million b/d. Russian export consists more than 5 million b/d of oil and nearly 2 million b/d of refined products, which go mainly to the Europe market. The domestic demand in 2005 was 2.6 million b/d in averaged. It is also the main transit country for oil from Kazakhstan.

The Russian oil industry is in need of huge investment. Strong growth in the Russian economy means that local demand for energy of all types (oil, gas, nuclear, coal, hydro, electricity) is continuing to grow.

Production

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Production

Import

Total Imports of Petroleum (Top 15 Countries)(Thousand Barrels per Day)

Country Apr-11 Mar-11 YTD 2011 Apr-10 YTD 2010

CANADA 2,625 2,666 2,736 2,486 2,523

SAUDI ARABIA 1,107 1,108 1,108 1,257 1,069

MEXICO 1,077 1,319 1,221 1,276 1,202

VENEZUELA 997 1,067 1,022 950 982

NIGERIA 922 913 955 1,125 1,009

RUSSIA 704 690 593 587 491

IRAQ 519 382 411 490 502

COLOMBIA 516 399 367 423 344

ALGERIA 466 500 484 464 470

ANGOLA 277 280 310 508 405

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Export

# Exporting Nation (2006) (103bbl/d) (103m3/d)

1 Saudi Arabia (OPEC) 8,651 1,376

2 Russia 1 6,565 1,044

3 Norway 1 2,542 404

4 Iran (OPEC) 2,519 401

5 United Arab Emirates (OPEC) 2,515 400

6 Venezuela (OPEC) 1 2,203 350

Oil Reserve

Summary of Reserve Data as of 2011

Country

Reserves [17] Production [18] Reserve life 1

109 bbl 109 m3 106 bbl/d 103 m3/d years

Venezuela 296.5 47.14 2.9 460 275

Saudi Arabia 264 42.0 9.7 1,540 75

Canada 175 27.8 2.1 330 188

Iran 151.2 24.04 3.5 560 112

Iraq 143.1 22.75 2.6 410 150

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Kuwait 101.5 16.14 2.6 410 110

United Arab Emirates 97.8 15.55 2.7 430 102

Russia 74.2 11.80 9.9 1,570 17

Libya 47 7.5 1.4 220 93

Nigeria 37 5.9 1.7 270 66

Saudi Arabia Petroleum

Energy in Saudi Arabia describes energy and electricity production, consumption and export in Saudi Arabia.

Saudi Arabia is one of the world's largest energy producers, pumping approximately 10.782 million barrels per day(1.7142×106 m 3 /d ) of petroleum. While most of this is exported, domestic use is rapidly increasing, primarily for electricity production.

Saudi Arabia's economy is petroleum-based; roughly 75% of government revenues and 90% of export earnings come from the oil industry. The oil industry produces about 45% of Saudi Arabia's gross domestic product, against 40% from the private sector. Saudi Arabia has per capita GDP of $20,700. The economy is still very dependent on oil in spite of a diversification effort, in particular in the petrochemical sector, with Sabic.

Production

Saudi Arabia produces slightly over 11,000,000 barrels (1,700,000 m3) of oil per day, exporting 8,900,000 barrels per day (1,415,000 m3/d). The government is investing over $71 Billion to increase oil production to near 12,000,000 barrels per day (1,900,000 m3/d) by 2009 and up to 12,500,000 barrels per day (1,990,000 m3/d) by 2015.[2] This may be attributed to the report that 700,000 barrels (111,000 m3) of excess capacity are needed to compensate for a natural decline in availability.

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Consumption

Import

Crude Oil Imports (Top 15 Countries)(Thousand Barrels per Day)

Country Apr-11 Mar-11 YTD 2011 Apr-10 YTD 2010

CANADA 2,079 2,151 2,142 1,883 1,921

SAUDI ARABIA 1,089 1,107 1,102 1,245 1,061

MEXICO 973 1,186 1,097 1,134 1,063

VENEZUELA 902 957 923 851 894

NIGERIA 856 840 902 1,092 982

IRAQ 519 382 411 490 502

COLOMBIA 462 363 328 364 309

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Export

# Exporting Nation (2006) (103bbl/d) (103m3/d)

1 Saudi Arabia (OPEC) 8,651 1,376

2 Russia 1 6,565 1,044

3 Norway 1 2,542 404

4 Iran (OPEC) 2,519 401

5 United Arab Emirates (OPEC) 2,515 400

Oil Reserve

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Further Exploration There are careers in Petroleum’s Industry that need a strong base in Additional Mathematics. To explore about this, we have interviewed an individual who has been involved in this industry for 23 years starting from 1988 till now. He who worked as an Offshore Installation Manager namely Mansor Shah bin Hassan is now a very hardship and knowledgeable 44 years old man.

From the interview, Mr. Mansor states that mathematics plays a very big role in every work of his. This include of usage of calculus in determining the amount of energy contain in a source of fuel.

Apart from that, he also mentioned other Additional Mathematics related career in petroleum industry such as petroleum engineer. The main work of this kind of engineer is to find the correct proportion of the petroleum content using differential calculus concept. The engineer must also master the mathematical programming as almost of the operation at the industry are conducted by the software in the computer.

From all this, we know that additional mathematics is needed in the petroleum industry. So, the understanding in this subject is very important to those who want to involve in this industry.

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ConclusionPetroleum, acting as a main source of energy in Malaysia, is a very significant remark of Malaysia’s richness of resources of land. We have depended on petroleum since more than 20 years. As a result of untiring works of Malaysia’s conglomerate, PETRONAS, our nation has been recognized as one of the top 30 oil producers.

The works that contributed a remarkable achievement that makes Malaysians proud actually began from the very tough works, weather in management or in tough labor operations. Neglecting the sweat of workers resulted from committing these rough tasks, one should also realize that these tasks are not only be done by physical means. Thanks to the development of scientific studies, the efficiency of energy usage has increase dramatically.

Not to forget the contribution of knowledge in Mathematics, especially Additional Mathematics, one could save much time of travelling from a venue to another by the application of geometry and algebra. For engineering, calculus, which includes the knowledge of differentiation and integration have contributed a lot in live operation at the oil tank and rigs.

To cope with the challenges in future, the nation demands a significant number of professionals in this field. In order to fulfill the need, students should involve themselves in sciences, preferably Mathematics. This is because Mathematics is the foundation of all sciences, like the saying goes “Mathematics is the queen of sciences”. The realization of this need will not only benefit the students nor the schools, but it will also means big to the nation, thus making Malaysia as a new, highly-developed country in the means of scientific values.

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