Question Answers for Additional Mathematics Project Work 2 2011 SCRIBD

8/4/2019 Question Answers for Additional Mathematics Project Work 2 2011 SCRIBD

1/25

[Pick the date]

[Type the document subtitle] | Assistant

MRSM

TUN

GHAFAR

BABA

ADDITIONAL MATHEMATICS PROJECTWORK 2/2011

NAME:

CLASS:

IC NO:

COLLEGE NUMBER:

TEACHERs N AME:


2/25

Question

Part I

Cakes come in a variety of forms and flavours and are among favourite desserts servedduring 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 andcake decorating. Find out how mathematics is used in cake baking and cake decoratingand write about your findings.

Answer:

Geometry To determine suitable dimensions for the cake, to assist in designing anddecorating cakes that comes in many attractive shapes and designs, to estimate volume ofcake to be produced

Calculus (differentiation) To determine minimum or maximum amount of ingredients for

cake-baking, to estimate minimum or maximum amount of cream needed for decorating, toestimate minimum or maximum size of cake produced.

Progressions To determine total weight/volume of multi-storey cakes with proportionaldimensions, to estimate total ingredients needed for cake-baking, to estimate total amountof cream for decoration.


3/25

Part II

Best Bakery shop received an order from your school to bake a 5 kg of round cake asshown in Diagram 1 for the Teachers Daycelebration.

(Diagram 11)

1) If a kilogram of cake has a volume of 3800 , and the height of the cake is to be7.0cm, calculate the diameter of the baking tray to be used to fit the 5 kg cakeordered by your school.

[Use = 3.142]

Answer:

Volume of 5kg cake = Base area of cake x Height of cake

3800 x 5 = (3.142)( ) x 7

(3.142) = ( )

863.872 = ( )

= 29.392

d = 58.784 cm


4/25

2) The cake will be baked in an oven with inner dimensions of 80.0 cm in length, 60.0 cm inwidth 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

Answer:

First, form the formula for d in terms of h by using the above formula for volume of cake, V= 19000, that is:

19000 = (3.142)(d/2)h

=

= d

d =

Height,h (cm) Diameter,d (cm)

1.0 155.53

2.0 109.98

3.0 89.80

4.0 77.77

5.0 68.56

6.0 63.49

7.0 58.78

8.0 54.99

9.0 51.8410.0 49.18


5/25

(b)Based on the values in your table,

(i) state the range of heights that is NOT suitable for the cakes and explain youranswers.

Answer :

h< 7cm is NOT suitable, because the resulting diameter produced is too large to fitinto the oven. Furthermore, the cake would be too short and too wide, making it lessattractive.

(ii) suggest the dimensions that you think most suitable for the cake. Give reasons

for your answer.

Answer:

h = 8cm, d = 54.99cm, because it can fit into the oven, and the size is suitable foreasy handling.


6/25

(c)

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

Answer:

19000 = (3.142)( )h

19000/(3.142)h =

= d

d =

d =

log d =

log d = log h + log 155.53

Log h 0 1 2 3 4Log d 2.19 1.69 1.19 0.69 0.19


7/25


8/25

(ii)

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

Answer:

h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm

(b) If Best Bakery used a 42 cm diameter round cake tray, use your graph to estimate theheight of the cake obtained.

Answer:

d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm

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

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

Answer:

h = 8cm, d = 54.99cmAmount of fresh cream = VOLUME of fresh cream needed (area x height)Amount of fresh cream = Volume of cream at the top surface + Volume of cream at theside surface

Volume of cream at the top surface= Area of top surface x Height of cream

= (3.142)( ) x 1

= 2375 cm

Volume of cream at the side surface= Area of side surface x Height of cream= (Circumference of cake x Height of cake) x Height of cream= 2(3.142)(54.99/2)(8) x 1= 1382.23 cm

Therefore, amount of fresh cream needed = 2375 + 1382.23 = 3757.23 cm


9/25

(b) Suggest three other shapes for cake, that will have the same height and volume asthose suggested in 2(b)(ii). Estimate the amount of fresh cream to be used on each of thecakes.

Answer:

1 Rectangle-shaped base (cuboid)

19000 = base area x height

base area =

length x width = 2375By trial and improvement, 2375 = 50 x 47.5 (length = 50, width = 47.5, height = 8)

Therefore, volume of cream

= 2(Area of left/right side surface)(Height of cream) + 2(Area of front/back sidesurface)(Height of cream) + Volume of top surface= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm


10/25

2 Triangle-shaped base

19000 = base area x heightbase area = 2375

x length x width = 2375

length x width = 4750

By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)Slant length of triangle = (95 + 25)= 98.23

Therefore, amount of cream= Area of rectangular front side surface(Height of cream) + 2(Area of slant rectangularleft/right side surface)(Height of cream) + Volume of top surface= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm


11/25

3 Pentagon-shaped base

19000 = base area x heightbase area = 2375 = area of 5 similar isosceles triangles in a pentagontherefore:2375 = 5(length x width)475 = length x widthBy trial and improvement, 475 = 25 x 19 (length = 25, width = 19)

Therefore, amount of cream= 5(area of one rectangular side surface)(height of cream) + volume of top surface= 5(8 x 19) + 2375 = 3135 cm

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

Answer:

Pentagon-shaped cake, since it requires only 3135 cm of cream to be used.


12/25

Part III

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

Answer:

Method 1: Differentiation

Use two equations for this method: the formula for volume of cake (as in Q2/a), and theformula for amount (volume) of cream to be used for the round cake (as in Q3/a).

19000 = (3.142)rh (1) V = (3.142)r + 2(3.142)rh (2)

From (1): h = (3)

Sub. (3) into (2):

V = (3.142)r + 2(3.142)r( )

V = (3.142)r + ( )

V = (3.142)r + 38000r -1

( ) = 2(3.142)r ( )

0 = 2(3.142)r ( ) -->> minimum value, therefore = 0

= 2(3.142)r

= r

6047.104 = rr = 18.22

Sub. r = 18.22 into (3):

h =

h = 18.22


13/25

therefore, h = 18.22cm, d = 2r = 2(18.22) = 36.44cm

Method 2: Quadratic Functions

Use the two same equations as in Method 1, but only the formula for amount of cream is

the main equation used as the quadratic function.Let f(r) = volume of cream, r = radius of round cake:

19000 = (3.142)rh (1) f(r) = (3.142)r + 2(3.142)hr (2) From (2):f(r) = (3.142)(r + 2hr) -->> factorize (3.142)

= (3.142)[ (r + ) ( ) ] -->> completing square, with a = (3.142), b = 2h and c = 0= (3.142)[ (r + h) h ]

= (3.142)(r + h) (3.142)h(a = (3.142) (positive indicates min. value), min. value = f(r) = (3.142)h, correspondingvalue of x = r = --h)

Sub. r = --h into (1):19000 = (3.142)(--h)hh = 6047.104h = 18.22

Sub. h = 18.22 into (1):19000 = (3.142)r(18.22)

r = 331.894r = 18.22

therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cm

I would choose not to bake a cake with such dimensions because its dimensions arenot suitable (the height is too high) and therefore less attractive. Furthermore, suchcakes are difficult to handle easily.


14/25

FURTHER EXPLORATION

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

The height of each cake is 6.0 cm and the radius of the largest cake is 31.0 cm. The radiusof the second cake is 10% less than the radius of the first cake, the radius of the third cakeis 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. Bycomparing all these values, determine whether the volumes of the cakes form anumber pattern?Explain and elaborate on the number patterns.

Answer:height, h of each cake = 6cm

radius of largest cake = 31cmradius of 2 nd cake = 10% smaller than 1 st cakeradius of 3 rd cake = 10% smaller than 2 nd cake

31, 27.9, 25.11, 22.599

a = 31, r =

V = (3.142)rh

Radius of 1 st cake = 31, volume of 1 st cake = (3.142)(31)(6) = 18116.772 Radius of 2 nd cake = 27.9, volume of 2 nd cake = 14674.585 Radius of 3 rd cake = 25.11, volume of 3 rd cake = 11886.414 Radius of 4 th cake = 22.599, volume of 4 th cake = 9627.995


15/25

18116.772, 14674.585, 11886.414, 9627.995, a = 18116.772, ratio, r = T 2/T1 = T3 /T2 = = 0.81

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

Answer :

S n =

S n = 57000, a = 18116.772 and r = 0.81

57000 =

1 0.81 n = 0.59779

0.40221 = 0.81 n

og 0.81 0.40221 = n

n =

n = 4.322

therefore, n 4


16/25

REFLECTION

I am glad that I was given the opportunity to take Additional Mathematics. It hasexpanded my knowledge in Mathematics. Despite it being a mere supplementary subject, Itook it seriously and managed to handle it well.

It has driven me to strive for the best in whatever I do, which is linked to our schoolmotto. Doing well in Additional Math has increased my confidence as it proves that I canachieve greater heights in my endeavours if I put my heart to it. To score marks inAdditional Mathematics, I did a lot of practices which needed patience.

I think I have improved in my skills and learnt how to think outside the box. I havealso learnt to encourage myself when I encounter any failure.

After spending countless hours, day and night to finish this Additional MathematicsProject, I realize how important Additional Mathematics is. Also, completing this projectmakes me realize how fun it is and likable is Additional Mathematics.

Do not worry about your difficulties with mathematics; I can assure you that mine aregreater still.

~ Albert Einstein


17/25

ACKNOWLEDGMENT

First of all, I would like to say Alhamdulillah, for giving me the strength andhealth to do this project work and finish it on time.

Not forgotten to my parents for providing everything, such as money, to buyanything that are related to this project work, their advise, which is the most neededfor this project and facilities such as internet, books, computers and all that. They alsosupported me and encouraged me to complete this task so that I will not procrastinatein doing it.

Then I would like to thank to my teacher, Cik Nurul Aishah binti Kamaruddin forguiding me throughout this project. Eventhough I had some difficulties in doing thistask, she taught me patiently until I knew what to do. She gave great guides andencouragement until I understand how to solve the project work.

Besides that,my appreaciation to my friends who always supporting me. Eventhis project is individually but we were helping each other through disscussion andsharing ideas to ensure our task will be done flawlessly.

Last but not least, I would like to thank any party which involved either directlyor indirect in completing this project work.

Thank you everyone.


18/25

INTRODUCTION

Geometry HistoryGeometry was thoroughly organized in about 300 BC, when the Greekmathematician Euclid gathered what was known at the time, added original work of hisown, and arranged 465 propositions into 13 books, called 'Elements'. The books coverednot only plane and solid geometry but also much of what is now known asalgebra,trigonometry, and advanced arithmetic.

Through the ages, the propositions have been rearranged, and many of the proofsare different, but the basic idea presented in the 'Elements' has not changed. In the workfacts are not just cataloged but are developed in a fashionable way.

Even in 300 BC, geometry was recognized to be not just for mathematicians.Anyone can benefit from the basic learning of geometry, which is how to follow lines ofreasoning, how to say precisely what is intended, and especially how to prove basicconcepts by following these lines of reasoning. Taking a course in geometry is beneficialfor all students, who will find that learning to reason and prove convincingly is necessaryfor every profession. It is true that not everyone must prove things, but everyone isexposed to proof. Politicians, advertisers, and many other people try to offer convincingarguments. Anyone who cannot tell a good proof from a bad one may easily be persuadedin the wrong direction. Geometry provides a simplified universe, where points and linesobey believable rules and where conclusions are easily verified. By first studying how toreason in this simplified universe, people can eventually, through practice and experience,

learn how to reason in a complicated world.Geometry in ancient times was recognized as part of everyone's education. Early

Greek philosophers asked that no one come to their schools that had not learned theElements' of Euclid. There were, and still are, many who resisted this kind of education.

Ancient knowledge of the sciences was often wrong and wholly unsatisfactory bymodern standards. However not all of the knowledge of the more learned peoples of thepast was false. In fact without people like Euclid or Plato we may not have been asadvanced in this age as we are.

Mathematics is an adventure in ideas. Within the history of mathematics, one findsthe ideas and lives of some of th e most brilliant people in the history of mankindspopulace upon Earth. First man created a number system of base 10. Certainly, it is not

just coincidence that man just so happens to have ten fingers or ten toes, for when ourprimitive ancestors first discovered the need to count they definitely would have used theirfingers to help them along just like a child today. When primitive man learned to count up toten he somehow differentiated him from other animals. As an object of a higher thinking,man invented ten number-sounds. The needs and possessions of primitive man were notmany. When the need to count over ten aroused, he simply combined the number-sounds


19/25

related with his fingers. So, if he wished to define one more than ten, he simply said one-ten. Thus our word eleven is simply a modern form of the Teutonic ein-lifon.

Since those first sounds were created, man has only added five new basic number-sounds to the ten primary ones. They are hundred, thousand, million, billion (athousand milli ons in America, a million millions in England), trillion (a million millions iAmerica, a million-million millions in England). Because primitive man invented the same

number of number-sounds as he had fingers, our number system is a decimal one, or ascale based on ten, consisting of limitless repetitions of the first ten number sounds.Undoubtedly, if nature had given man thirteen fingers instead of ten, our number systemwould be much changed.

For instance, with a base thirteen number system we would call fifteen, two-thirteens. While some intelligent and well-schooled scholars might argue whether or notbase ten is the most adequate number system, base ten is the irreversible favorite amongall the nations. Of course, primitive man most certainly did not realize the concept of thenumber system he had just created. Man simply used the number-sounds loosely asadjectives. So an amount of ten fish was ten fish, whereas ten is an adjective describingthe noun fish. Soon the need to keep tally on ones counting raised. The simple solutionwas to make a vertical mark. Thus, on many caves we see a number of marks that theresident used to keep track of his possessions such a fish or knives. This way of recordkeeping is still taught today in our schools under the name of tally marks.

The earliest continuous record of mathematical activity is from the secondmillennium BC when one of the few wonders of the world was created mathematics wasnecessary. Even the earliest Egyptian pyramid proved that the makers had a fundamentalknowledge of geometry and surveying skills. The approximate time period was 2900 BCThe first proof of mathematical activity in written form came about one thousand yearslater. The best-known sources of ancient Egyptian mathematics in written format are theRhind Papyrus and the Moscow Papyrus. The sources provide undeniable proof that thelater Egyptians had intermediate knowledge of the following mathematical problems,applications to surveying, salary distribution, calculation of area of simple geometricfigures' surfaces and volumes, simple solutions for first and second degree equations.Egyptians used a base ten number system most likely because of biologic reasons (tenfingers as explained above). They used the Natural Numbers (1,2,3,4,5,6, etc.) also knownas the counting numbers. The word digit, which is Latin for finger, is also another name fornumbers that explains the influence of fingers upon numbers once again.

The Egyptians produced a more complex system then the tally system for recordingamounts. Hieroglyphs stood for groups of tens, hundreds, and thousands. The higherpowers of ten made it much easier for the Egyptians to calculate into numbers as large asone million. Our number system which is both decimal and positional (52 is not the samevalue as 25) differed from the Egyptian, which was additive, but not positional. TheEgyptians also knew more of pi then its mere existence. They found pi to equal C/D or4(8/9) whereas a equals 2. The method for ancient peoples arriving at this numericalequation was fairly easy. They simply counted how many times a string that fit thecircumference of the circle fitted into the diameter, thus the rough approximation of 3. Thebiblical value of pi can be found in the Old Testament (I Kings vii.23 and 2 Chroniclesiv.2)in the following verse Also, he made a molten sea of ten cubits from brim to brim,round in compass, and five cubits the height thereof; and a line of thirty cubits did compass


20/25

it round about. The moltensea, as we are told is round, and measures thirty cubits roundabout (in circumference) and ten cubits from brim to brim (in diameter). Thus the biblicalvalue for pi is 30/10 = 3.

Now we travel to ancient Mesopotamia, home of the early Babylonians. Unlike theEgyptians, the Babylonians developed a flexible technique for dealing with fractions. The

Babylonians also succeeded in developing more sophisticated base ten arithmetic thatwere positional and they also stored mathematical records on clay tablets. Despite allThe real birth of modern math was in the era of Greece and Rome. Not only did thephilosophers ask the question how of previous cultures, but they also asked the modernquestion of why. The goal of this new thinking was to discover and understand the reasonfor mans existence in the universe and also to find his place. The philosophers of Greeceused mathematical formulas to prove propositions of mathematical properties. Some ofwho, like Aristotle, engaged in the theoretical study of logic and the analysis of correctreasoning. Up until this point in time, no previous culture had dealt with the negatedabstract side of mathematics, of with the concept of the mathematical proof.

The Greeks were interested not only in the application of mathematics but also in itsphilosophical significance, which was especially appreciated by Plato (429-348 BC). Platowas of the richer class of gentlemen of leisure. He, like others of his class, looked downupon the work of slaves and crafts worker. He sought relief, for the tiresome worries of life,in the study of philosophy and personal ethics. Within the walls of Platos academy at leastthree great mathematicians were taught, Theaetetus, known for the theory of irrational,Eodo xus, the theory of proportions, and also Archytas (I couldnt find what made him greatbut three books mentioned him so I will too). Indeed the motto of Platos academy Let noone ignorant of geometry enter within these walls was fitting for the scene of the greatminds who gathered here.

Another great mathematician of the Greeks was Pythagoras who provided one ofthe first mathematical proofs and discovered incommensurable magnitudes, or irrationalnumbers. The Pythagorean theorem relates the sides of a right triangle with theircorresponding squares. The discovery of irrational magnitudes had another consequencefor the Greeks since the length of diagonals of squares could not be expressed by rationalnumbers in the form of A over B, the Greek number system was inadequate for describingthem.

As, you might have realized, without the great minds of the past our mathematicalexperiences would be quite different from the way they are today.This, the greatest andmost remarkable feature of Babylonian Mathematics was their complex usage of asexagesimal place-valued system in addition a decimal system much like our own modernone. The Babylonians counted in both groups of ten and sixty. Because of the flexibility of asexagismal system with fractions, the Babylonians were strong in both algebra and numbertheory. Remaining clay tablets from the Babylonian records show solutions to first, second,and third degree equations. Also the calculations of compound interest, squares andsquare roots were apparent in the tablets. The sexagismal system of the Babylonians isstill commonly in usage today. Our system for telling time revolves around a sexagesimalsystem. The same system for telling time that is used today was also used by theBabylonians. Also, we use base sixty with circles (360 degrees to a circle). Usage of thesexagesimal system was principally for economic reasons. Being, the main units of weight


21/25

and money were mina,(60 shekels) and talent (60 mina). This sexagesimal arithmetic wasused in commerce and in astronomy.

The Babylonians used many of the more common cases of the PythagoreanTheorem for right triangles. They also used accurate formulas for solving the areas,volumes and other measurements of the easier geometric shapes as well as trapezoids.

The Babylonian value for pi was a very rounded off three. Because of this crudeapproximation of pi, the Babylonians achieved only rough estimates of the areas of circlesand other spherical, geometric objects.


22/25


23/25

OBJECTIVE ~

Apply mathematics to everyday situations and appreciate the importance ofmathematics in everyday life.

Improve problem-solving skills, thinking skills, reasoning and mathematicalcommunication.

Develop positive attitude and personalities and intrinsic mathematical values suchas accuracy, confidence and systemic reasoning.

Stimulate learning environment that enhances effective learning , inquiry-based andteam work.

Develop mathematical knowledge in a way which increases students interest and

confidence.


24/25


25/25

Documents

Question Answers for Additional Mathematics Project Work 2 2011 SCRIBD