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CURRICULUM DEVELOPMENTDIVISION MINISTRY OF

EDUCATION MALAYSIA

ADDITIONAL MATHEMATICSPROJECT WORK 2/2011

NAME: NOR NADIAH NADHIRAHBT MD NADZRI

CLASS : 5 SCIENCE 1TITLE : MATHEMATICS IN CAKE

BAKING AND CAKE

DECORATINGTEACHERS NAME: MRS NORAINIBT SALDANSCHOOL : SMS SULTAN MOHAMADJIWA


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CONTENT

NO CONTENT PAGE1 APPRECIATION

2 INTRODUCTION3 PART 14 PART 25 PART 36 FURTHER

EXPLORATION

7 REFLECTION


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APPRECIATION

Alhamdulillah, thank to God for giving me the will to

complete this Additional Mathematics project work.

Secondly, I would like to thank my Additional Mathematics

teacher, Mrs Noraini bt Saldan for her guide and gives a

lot of useful and important information for me to complete

this task. Besides that, I would like to appreciate my

parents for their support and encouragement throughout

the days for me to finish this project work. Lastly, a special

thank to all my friends for their help and cooperation in

searching and completing the task.


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INTRODUCTION

As students taking Additional Mathematics, we arerequired to carry out a project work while we are in Form5. This year, Curriculum Development Division Ministry ofEducation had prepared three tasks for us. We are need tochoose and complete ONE task only based on our areainterest. Upon completion of Additional MathematicsProject Work, we gain valuable experiences and able to:

Apply and adapt a variety of problem solvingstrategies to solve routine and non-routine problems.

Experience classroom environments which arechallenging, interesting and meaningful and hence,improving our thinking skills.

Experience classroom environments whereknowledge and skills are applied in meaningful waysin solving real life problems.

Acquire effective mathematical communicationthrough oral and writing , and to use the language ofmathematics to express mathematical ideas correctly

and precisely. Enhance acquisition of mathematical knowledge and

skills through problem solving in ways that increaseinterest and confidence.

Prepare ourselves for the demands of our futureundertakings and in workplace.

Realise that mathematics is an important andpowerful tool in solving real life problems and hencedevelop positive attitude towards mathematics.

Train ourselves to appreciate the intrinsic values ofmathematics and to become more creative.


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Realize the importance and beauty of mathematics.


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PART1

History of differentiation

The concept of a derivative in the sense of a tangent line is avery old one, familiar to Greek geometers such as Euclid (c. 300BC), Archimedes (c. 287212 BC) and Apollonius of Perga (c.262190 BC). Archimedes also introduced the use

ofinfinitesimals, although these were primarily used to study
http://en.wikipedia.org/wiki/Tangent_linehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Tangent_line


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areas and volumes rather than derivatives and tangents;seeArchimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found

in Indian mathematics, perhaps as early as 500 AD, when theastronomer and mathematician Aryabhata (476550) usedinfinitesimals to study the motion of the moon. The use ofinfinitesimals to compute rates of change was developedsignificantly by Bhskara II (1114-1185); indeed, it has beenargued that many of the key notions of differential calculus canbe found in his work, such as "Rolle's theorem".ThePersianmathematician, Sharaf al-Dn al-Ts(1135-1213), was the firstto discover the derivative ofcubic polynomials, an importantresult in differential calculus; his Treatise onEquations developed concepts related to differential calculus,such as the derivative function and the maxima and minima ofcurves, in order to solve cubic equations which may not havepositive solutions.

The modern development of calculus is usually credited to IsaacNewton (1643 1727) and Gottfried Leibniz (1646 1716), whoprovided independentand unified approaches to differentiationand derivatives. The key insight, however, that earned them thiscredit, was the fundamental theorem of calculus relatingdifferentiation and integration: this rendered obsolete mostprevious methods for computing areas and volumes, which hadnot been significantly extended since the time ofIbn al-Haytham (Alhazen). For their ideas on derivatives, both Newtonand Leibniz built on significant earlier work by mathematicianssuch as Isaac Barrow (1630 1677), Ren Descartes (1596

1650), Christiaan Huygens (1629 1695), Blaise Pascal (1623 1662) and John Wallis (1616 1703). Isaac Barrow is generalllygiven credit for the early development of thederivative. Nevertheless, Newton and Leibniz remain key figuresin the history of differentiation, not least because Newton wasthe first to apply differentiation to theoretical physics, whileLeibniz systematically developed much of the notation still usedtoday.
http://en.wikipedia.org/wiki/Archimedes'_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/Archimedes'_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Theoretical_physics


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Since the 17th century many mathematicians have contributedto the theory of differentiation. In the 19th century, calculus wasput on a much more rigorous footing by mathematicians suchas Augustin Louis Cauchy (1789 1857), Bernhard

Riemann (1826 1866), and Karl Weierstrass (1815 1897). Itwas also during this period that the differentiation wasgeneralized to Euclidean space and the complex plane.

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

cake-baking, to estimate minimum or maximum amount of

cream needed for decorating, to estimate minimum or maximumsize of cake produced.

Find the least area of metal required to make a closed cylindricalcontainer from thin sheet metal in order that it might have acapacity of 2000p cm3.

The total surface area of the cylinder, S, is 2pr2 + 2prhThe volume = pr2h = 2000p

Therefore pr2h = 2000p.Therefore h = 2000/r2

Therefore S = 2pr2 + 2pr( 2000/r2 )= 2pr2 + 4000p

r

So we have an expression for the surface area. To find when thesurface area is a minimum, we need to find dS/dr .

dS = 4pr - 4000pdr r2

When dS/dr = 0:4pr - (4000p)/r2 = 0Therefore 4pr = 4000p

r2

So 4pr3 = 4000pSo r3 = 1000So r = 10
http://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Complex_plane


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You should then check that this is indeed a minimum using thetechnique above.So the minimum area occurs when r = 10. This minimum area isfound by substituting into the equation for the area the value of r

= 10.

S = 2pr2 + 4000pr

= 2p(10)2 + 4000p10

= 200p + 400p= 600p

Therefore the minimum amount of metal required is 600p cm2

GeometryTo determine suitable dimensions for the cake, to assist in

designing and decorating cakes that comes in many attractiveshapes and designs, to estimate volume of cake to be producedFollowing are cakes in varies of geometry:


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PART2


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Part IIBest Bakery shop received an order from your school tobake a 5 kg of round cake as shownin 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 be 7.0cm, calculate the

diameter of the baking tray to be used to fit the 5 kgcake ordered 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


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2)The cake will be baked in an oven with inner dimensionsof 80.0 cm in length, 60.0 cmin width and 45.0 cm inheight.a)If the volume of cake remains the same, explore byusing different values of heights,hcm, and thecorresponding values of diameters of the baking tray tobeused,d cm. Tabulate your answers

Answer:First, form the formulaford in terms ofh by using theabove formula for volume of cake, V = 19000, that is:

19000 = (3.142)(d/2)h

=

= d

d =

Height,h (cm) Diameter,d(cm)1.0 155.532.0 109.983.0 89.804.0 77.775.0 68.566.0 63.497.0 58.788.0 54.999.0 51.84

10.0 49.18

(b)Based on the values in your table,


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(i)state the range of heights that isNOTsuitable for thecakes andexplain your answers.Answer:h< 7cm is NOT suitable, because the resulting

diameter produced is too large to fit into the oven.Furthermore, the cake would be too short and toowide, making it less attractive.

(ii)suggest the dimensions that you think most suitable forthe cake. Givereasons for your answer.Answer:h = 8cm, d = 54.99cm, because it can fit into the oven,and the size is suitable for easy handling.

(c)(i) Form an equation to represent the linear relationbetweenhand d. Hence, plot a suitable graph based on theequation that you haveformed. [You may draw your graphwith the aid of computersoftware.]

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


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(ii)(a) If Best Bakery received an order to bake a cake wherethe height of the cake is 10.5 cm, use your graph todetermine 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 yourgraph to estimate the height of the cake obtained.Answer:


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d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm3)Best Bakery has been requested to decorate the cakewith fresh cream. The thicknessof the cream is normallyset to a uniform layer of about1cm(a)Estimate the amount of fresh cream required todecorate the cake using the dimensions that you havesuggested 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 = Vol. of cream at the top surface+ Vol. of cream at the side surface

Vol. of cream at the top surface= Area of top surface x Height of cream

= (3.142)( ) x 1

= 2375 cm

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

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

(b)Suggestthreeother shapes for cake, that will have thesame height andvolume as those suggested in 2(b)(ii).Estimate the amount of fresh cream tobe used on each ofthe cakes.

Answer:


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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 side surface)(Height of cream) + Vol.

of top surface= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm

2 Triangle-shaped base

19000 = base area x heightbase area = 2375x length x width = 2375

length x width = 4750By trial and improvement, 4750 = 95 x 50 (length = 95,

width = 50)Slant length of triangle = (95 + 25)= 98.23


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PART3


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Part III

Find the dimension of a 5 kg round cake that requires the

minimum amount of fresh cream todecorate. Use at leasttwo different methods including Calculus.State whetheryou would choose to bake a cake of such dimensions.Give reasons for youranswers.

Answer:

Method 1: DifferentiationUse two equations for this method: the formula for volumeof cake (as in Q2/a), and the formula 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


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= r

6047.104 = rr = 18.22

Sub. r = 18.22 into (3):

h =

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

Method 2: Quadratic FunctionsUse the two same equations as in Method 1, but only theformula for amount of cream is the main equation used asthe 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, corresponding value 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


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

FURTHEREXPLORATI

ON&


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CONCLUSION

FURTHER EXPLORATIONBest Bakery received an order to bake a multi-storey cakefor Merdeka Day celebration, as shown in Diagram 2.The height of each cake is 6.0 cm and the radius of thelargest cake is 31.0 cm. The radius of the second cake is10% less than the radius of the first cake, the radius of thethird cake is10% less than the radius of the second cakeand so on.(a)Find the volume of the first, the second, the third and the

fourth cakes. By comparingall these values, determinewhether the volumes of the cakes form a number pattern?Explain and elaborate on the number patterns.Answer:

height, h of each cake = 6cm


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

31, 27.9, 25.11, 22.599

a = 31, r =

V = (3.142)rhRadius of 1st cake = 31, volume of 1st cake = (3.142)(31)(6) = 18116.772Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585

Radius of 3rd

cake = 25.11, vol. of 3rd

cake = 11886.414Radius of 4th cake = 22.599, vol. of 4th cake = 9627.995

18116.772, 14674.585, 11886.414, 9627.995,

a = 18116.772, ratio, r = T2/T1 = T3/T2 = = 0.81

(b) If the total mass of all the cakes shouldnot exceed 15 kg, calculate the maximumnumber of cakesthat the bakery needs to bake. Verify your answer usingothermethods.Answer:

Sn =

Sn = 57000, a = 18116.772 and r = 0.81

57000 =

1 0.81n = 0.59779

0.40221 = 0.81n

og0.81 0.40221 = n


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n =

n = 4.322

therefore, n 4

CONCLUSIONS

In cake baking and decoration, without myrealize, mathematics are commonly used. I saw theusage of geometry, differentiation and also

progression. By using geometry, the cakes becomemore attractive and adorable. While,differentiation and progression help us to estimatethe amount of ingredients that are needed whetherin baking or decorating. Without mathematics,one may bake a cake that has very high or verywide dimensions. This will cause problems tohimself to handle with those cakes.


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So, as I doing this project, I noticed thatgeometry, differentiation and progression can beclose in our daily life. Different shapes havedifferent volumes and amount of fresh cream to belayered. From differentiation, we can calculate themaximum or minimum value of fresh cream to beused correctly.

After we know and learn concept of thismathematics, we can apply them in our life. Inorder to bake a handy and nice cake, we cancapable to decide on shapes that are morefavourable and suitable.


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REFLECTION

REFLECTION.

While I conducting this project, a lot of informationthat I had found. I had learnt how to bake a cake with thebest dimensions and able to estimate the amount of freshcream needed to decorate that cake.

A part from that, this project encourage students towork together and share their knowledge. It is also


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encourage students to gather information from internet,improve thinking skills and promote effectivemathematical communication.

Last but not least, I proposed this project should be

continue because it brings a lot of moral values to thestudents understanding in Additional Mathematics.In the making of this project, I have spent countlesshours doing this project.I realized that this subjectis a compulsory to me. Without it, I cant fulfill mybig dreams and wishes.I used to hate Additional MathematicsIt always makes me wonder why this subject is sodifficult

I always tried to love every part of itIt always an absolute obstacle for meThroughout day and nightI sacrificed my precious time to have funFrom..Monday,Tuesday,Wednesday,Thursday,FridayAnd even the weekend that I always lookingforward to

From now on, I will do my best on everysecond that I will learn Additional

Mathematics.

Date post:	07-Apr-2018
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