Solution for SPM Add Math Project 2011 - Work 2 is available at forum.myhometuition.com now.

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[The solution will be available soon. Stay tune.]
Part 1
Rene Descartes, a renowned French Mathematician in the 16th century, discovered the beauty of Cartesian coordinates system while lying on his back and gazing at a spider on the ceiling. Do some research and write about his discoveries.

Part 2
Malaysia with its warm tropical climate is rich in flora and fauna. Beautiful gardens are found all over Malaysia. SMK Pennata decided to beautify the school compound by getting the students involved in the planting and maintenance of the greenery in the school compound as shown in Diagram 1. Each society is allocated a plot of land in various shapes and sizes to nurture throughout the year. The Mathematics Society, Englisl, Language Society and Malay Language Society are allocated the region P, Q and R respectively as shown in Diagram 1.

(a) Determine the area of region P, Q and R by using at least three different methods including the use of calculus. Ve -ify the answers obtained by using computer software. (Suggestions: GeoGebra, GSP, graphing calculator etc)

(b) Suppose there is a hcdgc along AB. The Mathematics Society wishes to fence up the remaining sides of the region P. Determine the length of fence required.

(c) If a meter of fence costs RM25.00, what is the total cost required by the Mathematics Society to fence up region P? Is it possible for the society to carry out the fencing with an allocation of RM250.00? Explain your answer.

(d) During the Mathematics Week, the society was given a single flag chain of length 9.20 meters to be used completely. The President of the society wishes to tie the flag chain continuously from A to E and then to another point along the hedge AB to create a triangular-shaped area.
(i) Make a conjecture about the number of points that the flag chain can be tied to along AB.
Support your conjecture with suitable calculations. Explain your answer.
(ii) Calculate the maximum area of the triangle obtained. Discuss.

Part 3
The Mathematics society decided to build a pond in region P as shown in Diagram 2. The pond is in the shape of a sector with centre E, radius ED and a depth of 1 meter.
(a) Calculate the angle AED, in radians, by using at least two different methods.
(b) Determine the volume of water that has to be pumped in to fill up 80% of the pond.
(c) If the water is pumped into the pond at a constant rate of 0.001 m3 s-', calculate
(i) the i ute of change of depth of the water,
(ii) the depth of water after 10 minutes,
(iii) the minimum time taken, in minutes, before the water overflows, and
(iv) the minimum time taken, in minutes, before the water overflows, if the pond is triangular-shaped AED and has a depth of 2 meters.

FURTHER EXPLORATION
Maps have been used for thousands of years to aid travelers during their journey from one place to another. Maps can also be used to estimate distance between places. In the year 2014, a recreation park will be constructed in town marked `X' on the map of Malaysia in as shown Diagram 3. This town has the latitude of 5° 41' N has the same longitude as the city of Malacca.

Explore and find the distance between these two places in kilometer by using,
(i) the map in Diagram 3
(ii) the formula given below:
Distance = θ x 60 nautical miles

where θ= difference in latitudes in degrees.
(a) Surf the Internet and use the Google map to locate the position of your school and two nearby hospitals/clinics. Print a copy of this Google map anc' mark the position of these three places.
(i) Solve the triangle obtained.
(ii) Calculate the shortest distance from your school to the line joining the two hospitals/clinics.

If Internet service is not available, you can perform this task by using a detailed map of your town.

REFLECTION
While you were conducting the project, what have you learnt? What moral values did you practise? Represent your opinions or feelings creatively through usage of symbols, illustrations. drawing or even in a song.

[The solution is available at forum.myhometuition.com now]

Part I

Cakes cone in a variety of forms and flavours and are among favourite desserts served during special occasions such as birthday parties, Hari Raya, weddings and etc. Cakes are treasured not only because of their wonderful taste but also in the art of cake baking and cake decorating. Find out how, mathematics is used in cake baking and cake decorating and write about your findings.

Part II
Best Bakery shop received an order from your school to bake a 5 kg of round cake as shown in Diagram 1 for the Teachers' Day celebration.

1) If a kilogram of cake has a volume of 3800cm³, and the height of the cake is to be 7.0 cm, calculate the diameter of the baking tray to be used to fit the 5 kg cake ordered by your school. [Use π = 3.142]

2) The cake will be baked in an oven with inner dimensions of 80.0 cm in length, 60.0 cm in width and 45.0 cm in height.

a) If the volume of cake remains the same, explore by using different values of heights, h cm, and the corresponding values of diameters of the baking tray to be used, d cm. Tabulate your answers.
(b) Based on the values in your table,
(i) state the range of heights that is NOT sui table for the cakes and explain your answers.
(ii) suggest the dimensions that you think most suitable for the cake. Give reasons for your answer.

(c) (i) Form an equation to represent the linear relation between h and d. Hence, plot a suitable graph based on the equation that you have formed. [You may draw your graph with the aid of computer software.]

(ii) (a) If Best Bakery received an order to bake a cake where the height of the cake is 10.5 cm, use your graph to determine the diameter of the round cake pan required.
(b) If Best Bakery used a 42 cm diameter round cake tray, use your graph to estimate the height of the cake obtained.

3) Best Bakery has been requested to decorate the cake with fresh cream. The thickness of the cream is normally set to a uniform layer of about 1 cm.

(a) Estimate the amount of fresh cream required to decorate the cake using the dimensions that you have suggested in 2(b)(ii).

(b) Suggest thr ee of her shapes fo r cake , that will have t he same height and volume as those suggested in 2(b)(ii). Estimate the amount of fresh cream to be used on each of the cakes.

(c) Based on the values that you have found which shape requires the least amount of fresh cream to be used?
Part III

Find the dimension of a 5 kg round cake that requires the minimum amount of fresh cream to decorate. Use at least two different methods including Calculus.
State whether you would choose to bake a cake of such dimensions. Give reasons for your answers.

FURTHER EXPLORATION

Best Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, as shown in Diagram 2.

The height of each cake is 6.0 cm and the radius of the largest cake is 31.0 cm. The radius of the second cake is 10% less than the radius of the first cake, the radius of the third cake is 10% less than the radius of the second cake and so on.

(a) Find the volume of the first, the second, the third and the fourth cakes. By comparing all these values, determine whether the volumes of the cakes form a number pattern? Explain and elaborate on the number patterns.

(b) If the total mass of all the cakes should not exceed 15 kg, calculate the maximum number of cakes that the bakery needs to bake. Verify your answer using other methods.

REFLECTION

While you were conducting the project, what have you learnt? What moral values did you practise? Represent your opinions or feelings creatively through usage of symbols, illustrations, drawing or even in a song.

[The solution will be available soon. Stay tune.]
Malaysia's Petronas finds new energy fields offshore Sarawak

Reuters - Monday, February 14, 2011

KUALA LUMPUR, Feb 14 - Malaysia's state oil firm Petronas said on Monday it has discovered new oil and gas fields through the drilling of NC3 and Spaoh-1 wells in Blocks SK316 and SK306 in offshore Sarawak state.

It said drilling of the NC3 wildcat well and a subsequent appraisal well brought significant discovery in Block SK316 with early estimation of 2.6 trillion standard cubic feet of net gas in place.

The Spaoh-1 well of 3,000 meters drilling depth, located in Block SK306 was drilled in December 2010 and an early evaluation showed around 100 million barrels of oil and 0.2 trillion standard cubic feet of gas in place, respectively, it said.

"In the next three years, over 50 exploration wells1 are expected to be drilled offshore Malaysia by Petronas and its production sharing contractors," Petronas said.

"These activities. especially if they result in discoveries, are expected to spur business opportunities in the oil and gas industry and will promote upstream investment in the country."

Oil production plays a very significant role in Malaysia's economic development. Information above shows the discovery of new oil reserves in Malaysia. Imagine that you are the leader of a team of professionals, and received a directive to go to the newly installed off-shore oil rig to perform some tasks. On the way to the oil rig, your team needs to stop by and does some maintenance job at an offshore wind-farm which provides the energy needs of the oil rig. The maintenance job takes 2 hours to complete.

Diagram 1 is given to facilitate your task. The arrows show the suggested route. The locations of sunken treasures and sunken ships are preserved as high security areas which are out of bound to any approaching ships. The coral reef is the ecologically sensitive area with many endangered animal and plant species which are protected by the Malaysian Fisheries Act 1985 (Act 317).

Notes:
1 A1 ildcat wells arc those drilled outside ()land not in the vicinity of knov. n oil or gas fields.
2. Appraisal wells arc used to assess characteristics (such as (loo rate) ul proven hydrocarbon accumulation.
3. Exploration wells are drilled purely for exploratory (information gathering) purposes in new areas.

PART 1
Suggest two other possible routes and state which route is the best.
Construct a table of comparison between the suggested route and the two other possible routes, Your table should include:
  • Distance
  • Bearings
  • Coordinates where the course of the boat needs to be changed
  • Possible dangers and possibilities of intruding into the preserved and conservation areas
  • Time estimation, assuming that the average speed of your boat is 35 km/hr.
Hence, justify your choice.

PART 2
Your team decides to use the suggested route as shown in Diagram 1 and starts the journey at 10:00 a.m. The team has to reach the oil rig to attend a meeting at 2.00 p.m. on the same day. Keep in mind that your team needs to stay 2 hours on the offshore wind-farm to check the wind generation system. There is a sea current flowing from east to west at a constant speed of 15 km/h. Your boat can travel at a speed of 36 km/h in still water. Determine the time your team reaches the wind-farm and the oil rig.

PART 3
Given that the power generated by a single wind turbine as shown in Diagram 2 is proportional to the square of the wind speed u, the surface area of the blade of the wind turbine A and the machine constant c, i.e. P = cAu².

The energy E produced in t seconds is given by


If your team have obtained the following measurements:
Wind speed = 13 m/s
Surface area of the wind turbine = 10 m²
Power output = 10 kW
(a) Find the value of the constant c.
(b) Determine how much time is required to generate 50 MJ of energy if
(i) the wind speed is constant at 13 m/s,
(ii) the function of the wind speed is given by u(t) = 0. 02t.

PART 4
Referring to the information given in the news from Reuters on 14 Feb 2011, `The Spaoh-1 well of 3,000 meters drilling depth, located iii Block SK306 was drilled in December 2010 and an early evaluation showed around 100 million barrels of oil...', calculate:

(a) the average rate of oil production (in barrels of oil per hour) for the oil well so that the oil supply could last for 10 years;

(b) the rate of increase in oil level in the barrel, in cm/min, if given that the barrel is a cylinder with diameter of 0.5 meter and height of 1 meters, and the time taken to fill up a barrel with oil is 5 minutes.

PART 5
Find information on the:
(a) oil production and consumption;
(b) oil imports and exports; and
(c) oil reserves

for major countries around the world. Write a brief report based on your findings. Your report must include suitable statistical representations.

FURTHER EXPLORATION
Gather further information regarding careers related to the Petroleum Industry which requires a good foundation in Additional Mathematics by:
(a) interviewing individuals who are in the Petroleum industry, or
(b) surfing the Internet.

Write a report based on your findings.

SPM Add Math Project 2011


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