Answers for Additional Mathematics Project Work 2 2011

8/6/2019 Answers for Additional Mathematics Project Work 2 2011

1/42

Additional Mathematics Project Work 2 2011

Page | 1

-@feris addmath work@-


2/42


Page | 2


First of all, I would like to say Alhamdulillah, for giving me the strength and health to

do this project work.

Not forgotten my parents for providing everything, such as money, to buy anything

that are related to this project work and their advise, which is the most needed for this

project. Internet, books, computers and all that. They also supported me and encouraged

me tocomplete this task so that I will not procrastinate in doing it.

Then I would like to thank my teacher, Pn. Hafizahwati bt for guiding me

and my friends throughout this project. We had some difficulties in doing this task, but he

taught us patiently until we knew what to do. He tried and tried to teach us until we

understand what we supposed to do with the project work.

Last but not least, my friends who were doing this project with me and sharing our

ideas. They were helpful that when we combined and discussed together, we had this task

done.


3/42


Page | 3



4/42


Page | 4


We students taking Additional Mathematics are required to carry out a project work while we

are in Form 5.This yearthe Curriculum Development Division, Ministry ofEducation has

prepared four tasks for us.We are to chooseand complete only ONE task based on our area

ofinterest.This project can be done in groups or individually,and I gladly choose to do this

individually.Upon completionof the Additional Mathematics Project Work,we are to

gainvaluable experiences and able to :

y Apply and adapt a variety of problem solving strategies to solve routine and non-routine

problems

y Experience classroom environments which are challenging, interesting and meaningful

and hence improve their thinking skills

y Experience classroom environments where knowledge and skills are applied in

meaningful ways in solving real-life problems.

y Experience classroom environments where expressing ones mathematical

thinking,reasoning and communication are highly encouraged and expected

y Experience classroom environments that stimulates and enhances effective learning.

y Acquire effective mathematical communication through oral and writing,and to use the

language of mathematics to express mathematical ideas correctly and precisely


5/42


Page | 5


y Enhance acquisition of mathematical knowledge and skills through problem-solving

in ways that increase interest and confidence

y Prepare ourselves for the demand of our future undertakings and in workplace

y Realise that mathematics is an important andpowerful tool in solving real-life problems

andhence develop positive attitude towards mathematics

y Train ourselves not only to be independent learners but also to collaborate, to cooperate,

and to share knowledge in an engaging and healthy environment


6/42


Page | 6



7/42


Page | 7


The aims of carrying out this project work are:

y to apply and adapt a variety of problem-solving strategies to solve problems

y to improve thinking skills

y to promote effective mathematical communication

y to develop mathematical knowledge through problem solving in a way that

increases students interest and confidence

y to use the language of mathematics to express mathematical ideas precisely

y to provide learning environment that stimulates and enhances effective learning

y to develop positive attitude towards mathematics

There are a lot of things around us related to circles or parts of a circle. A circle is asimple shape of Euclidean geometry consisting of those points in a plane which is thesame distance from a given point called the centre. The common distance of the points of acircle from its center is called its radius.

Circles are simple closed curves which divide the plane into two regions,

an interior and an exterior. In everyday use, the term "circle" may be used interchangeablyto refer to either the boundary of the figure (known as the perimeter) or to thewhole figure including its interior. However, in strict technical usage, "circle"refers to the perimeter while the interior of the circle is called a disk. The circumference ofa circle is the perimeter of the circle (especially when referring to its length).


8/42


Page | 8


A circle is a special ellipse in which the two foci are coincident. Circles are conicsections attained when a right circular cone is intersected with a plane perpendicular to theaxis of the cone.

The circle has been known since before the beginning of recorded history. It is thebasis for the wheel, which, with related inventions such as gears, makes much of moderncivilization possible. In mathematics, the study of the circle has helped inspire thedevelopment of geometry and calculus. Circles had been used in daily lives to help peoplein their living.


9/42


Page | 9



10/42


Page | 10


Part I

Identifying the Problem: Question asks to provide information about how Mathematics isused in baking and cake decorating.Strategy: Apply knowledge obtained from Additional Mathematics and also consultresources from the Internet.

Part IIQuestion 1Identifying the Problem: Question asks to find the diameter of the baking tray to be usedto fit the cake.Strategy: Find diameter, d using the formula of volume of cylinder, with the height, h andvolume, v of cake per kilogram and the value of.

Question 2(a)Identifying the Problem: Question asks to find and tabulate the different values heights anddiameters of the baking tray to be used, if the volume remains the same.Strategy: Find diameter, d using the formula of volume of cylinder and volume, v of cake perkilogram and the value of with varying values of height, h.

Question 2(b)(i)Identifying the Problem: To state the range of heights unsuitable for baking of cakes andgive reasons.Strategy: Obtain the range of heights not suitable for cakes by compare and contrast withlogic opinion.

Question 2(b)(ii)Identifying the Problem: Question asks to suggest the dimension most suitable for thecake, and provide reasons.Strategy: Compare and contrast and the heights and diameters in table from Question 2(a)and suggest logical reasons.


11/42


Page | 11


Question 2(c)(i)Identifying the Problem: Question asks to reduce the equation to linear form in order todisplay the relation between h and d. After doing so, question requires us to plot a graphbased on the equation.Strategy: Using the linear law, and the linear equation, reduce the equation in Question2(a) by using logarithm and plot a graph of h against d using software.

Question 2(c)(ii)(a)Identifying the Problem: Question asks to determine the diameter of the round cake panif the height of the cake is 10.5 cm using the graph.Strategy: Express the height of the cake using logarithms Base 10. Then plot the graph toobtain the value of diameter, d in logarithmic terms. Finally, express the value without

logarithms.

Question 2(c)(ii)(b)Identifying the Problem: Question asks to determine the height of the cake, h obtained ifthe diameter of the round tray is 42 cm.Strategy: Express the diameter of the baking tray, d in logarithms Base 10. Then plot thepoint on the graph in order to obtain the logarithm value of the height. Finally, express thevalue in normal terms.

Question 3(a)

Identifying the Problem: Question asks to calculate the amount of fresh cream required todecorate the cake based on the dimensions suggested in Question 2(b)(ii).Strategy: Using h = 11 cm and d = 46.89293 cm, and the given thickness of the cream = 1cm uniformly, calculate the amount of fresh cream required.

Question 3(b)Identifying the Problem: Question asks to give 3 other suggestions for cake shaped whichhas the same height and volume as 2(b)(ii), and then calculate the amount of fresh cream to beused.Strategy: Provide 3 other cake shape suggestions by logic thinking and calculate the

dimensions, given the volume and height are constant. After that, calculate the amount of freshcream required for each shape.


12/42


Page | 12


Part IIIIdentifying the Problem: Question asks to find the dimensions of cake that requires the

minimum amount of fresh cream to decorate by using 2 different method, includingcalculus. After that, to state whether if such dimensions are suitable for the baking of thecake and give reasons.Strategy: Find the dimension that requires the minimum amount of fresh cream bycomparing values of height against volume of cream used. The second method would beusing differentiation. Finally, apply logic thinking to come to a conclusion, whether the cakeis suitable to be baked or not.

Further Exploration (a)Identifying the Problem: Question asks to obtain the volume of the first, second, third,and fourth cakes, and compare the values to determine whether the volumes form anumber pattern. After that, provide explanation and elaboration.Strategy: Given height of cake is 6.0 cm each and the radius of the largest cake is 31.0cm, after that the radius of the second cake is 10% less, the third radius is 10% less thanthe second cake and so on. Use the information given to obtain the volumes of the cakes,and compare by division to determine the existence of the number pattern.

Further Exploration (b)Identifying the Problem: Question asks to calculate the maximum number of cakes to be

baked by the bakery if the total mass must not exceed 15 kg. After that, verify the answerusing other methods.Strategy: Express the mass of the cake given 1kg of cake has a volume of 3800 cm3.Then find the number of terms using the formula to given the sum is a geometricprogression. After that, verify the answer by trial and improvement to prove the number ofcakes that the bakery needs to bake.


13/42


Page | 13



14/42


Page | 14


History of Cake Baking and Decorating

Although clear examples of the difference between cake and bread are easy to find, the

precise classification has always been elusive. For example, banana bread may beproperly considered either a quick bread or a cake.The Greeks invented beer as aleavener, frying fritters in olive oil, and cheesecakes using goat's milk. In ancient Rome,basic bread dough was sometimes enriched with butter, eggs, and honey, which produceda sweet and cake-like baked good. Latin poet Ovid refers to the birthday of him and hisbrother with party and cake in his first book of exile, Tristia.Early cakes in England werealso essentially bread: the most obvious differences between a "cake" and "bread" werethe round, flat shape of the cakes, and the cooking method, which turned cakes over oncewhile cooking, while bread was left upright throughout the baking process. Sponge cakes,leavened with beaten eggs, originated during the Renaissance, possibly in Spain.

Cake decorating is one of the sugar arts requiring mathematics that uses icing or frostingand other edible decorative elements to make otherwise plain cakes more visuallyinteresting. Alternatively, cakes can be moulded and sculpted to resemble three-dimensional persons, places and things. In many areas of the world, decorated cakes areoften a focal point of a special celebration such as a birthday, graduation, bridal shower,wedding, or anniversary.

Mathematics are often used to bake and decorate cakes, especially in the followingactions:

y Measurement of Ingredientsy

Calculation of Price and Estimated Costy Estimation of Dimensionsy Calculation of Baking Timesy Modification of Recipe according to scale


15/42


Page | 15



16/42


Page | 16


Question

Cakes come in a variety of forms and flavours and are amongfavourite desserts servedduring special occasions such asbirthday parties, Hari Raya, weddings and etc. Cakesaretreasured not only because of their wonderful taste but also inthe art of cake baking andcake decorating. Find out howmathematics is used in cake baking and cake decoratingandwrite 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

When making a batch of cake batter, you end up with a certain volume, determined by the

recipe.

The baker must then choose the appropriate size and shape of pan to achieve the desired

result. If the pan is too big, the cake becomes too short. If the pan is too small, the cakebecomes too tall. This leads into the next situation.

The ratio of the surface area to the volume determines how much crust a baked good will

have. The more surface area there is, compared to the volume, the faster the item will

bake, and the less "inside" there will be. For a very large, thick item, it will take a long time

for the heat to penetrate to the center. To avoid having a rock-hard outside in this case, the

baker will have to lower the temperature a little bit and bake for a longer time.

We mix ingredients in round bowls because cubes would have corners where unmixed

ingredients would accumulate, and we would have a hard time scraping them into the

batter.


17/42


Page | 17


CALCULUS (DIFFERENTIATION)-To determine minimum or maximum amount of

ingredients for cake-baking, to estimate min. or max. amount of cream needed fordecorating, to estimate min. or max. Size of cake produced.

PROGRESSION-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.

For example when we make a cake with many layers, we must fix the difference of

diameter of the two layers. So we can say that it used arithmetic progression. When thediameter of the first layer of the cake is 8 and the diameter of second layer of the cake is

6, then the diameter of the third layer should be 4.

In this case, we use arithmetic progression where the difference of the diameter is constant

that is 2. When the diameter decreases, the weight also decreases. That is the way how

the cake is balance to prevent it from smooch. We can also use ratio, because when we

prepare the ingredient for each layer of the cake, we need to decrease its ratio from lower

layer to upper layer. When we cut the cake, we can use fraction to devide the cakeaccording to the total people that will eat the cake.


18/42


Page | 18



19/42


Page | 19


Best Bakery shop received an order from your school to bake 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 kg cake ordered by yourschool.

[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


20/42


Page | 20


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

width and 45.0 cm in height.a)If the volume of cake remains the same, explore by using different values ofheights,hcm, and the corresponding values of diameters of the baking tray tobeused,d cm. Tabulate your answers

Answer:

First, form the formulaford in terms ofh 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 68.566.0 63.49

7.0 58.78

8.0 54.99

9.0 51.84

10.0 49.18


21/42


Page | 21


(b)Based on the values in your table,

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

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. 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.


22/42


Page | 22


(c)

(i) Form an equation to represent the linear relation betweenhand d. Hence,plot a suitable graph based on the equation that you haveformed. [You maydraw your graph with 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


23/42


Page | 23



24/42


Page | 24


(ii)

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

Answer:

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


25/42


Page | 25


3)Best Bakery has been requested to decorate the cake with fresh cream. The thicknessof

the cream is normally set to a uniform layer of about1cm(a)Estimate the amount of fresh cream required to decorate the cake usingthedimensions that 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 = Vol. of cream at the top surface + Vol. of cream at the sidesurface

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 of cream= 2(3.142)(54.99/2)(8) x 1= 1382.23 cm

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


26/42


Page | 26


(b)Suggest three other shapes for cake, that will have the same height and volume

as those suggested in 2(b)(ii). Estimate the amount of fresh cream tobe used oneach of the cakes.

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


27/42


Page | 27


2 Triangle-shaped base

19000 = base area x heightbase area = 2375

x 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

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


28/42


Page | 28


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) + vol. of top surface= 5(8 x 19) + 2375 = 3135 cm

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

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


29/42


Page | 29



30/42


Page | 30


Find the dimension of a 5 kg round cake that requires the minimum amount of fresh creamtodecorate. Use at least two different methods including Calculus.State whether you would

choose to bake a cake of such dimensions. Give reasons for youranswers.

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


31/42


Page | 31


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, 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

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.


32/42


Page | 32



33/42


Page | 33


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 cakeis10% 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 comparingallthese values, determine whether the volumes of the cakes form a number pattern?Explainand elaborate on the number patterns.


34/42


Page | 34


Answer:

height, h of each cake = 6cm

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 = 14674.585Radius 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


35/42


Page | 35


(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 answerusing othermethods.

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

n =

n = 4.322

therefore, n 4


36/42


Page | 36



37/42


Page | 37


After spending countless hours,days and night to finish thisproject and also sacrificing my time video games and mangas

in this mid year holiday,there are several things that I cansay...

Additional Mathematics...From the day I born...From the day I was able to holding pencil...From the day I start learning...

And...From the day I heard your name...I always thought that you will be my greatest obstacle andrival in excelling in my life...But after countless of hours...Countless of days...Countless of nights...

After sacrificing my precious time just for you...Sacrificing my Computer Games...Sacrificing my Video Games...Sacrificing my Facebook...

Sacrificing my Internet...Sacrifing my Anime...Sacrificing my Manga...Yahoo MassengerI realized something really important in you...I really love you...You are my real friend...You my partner...You are my soulmate...I LOVE U ADDITIONAL MATHEMATICS...


38/42


Page | 38


TEAM WORK IS IMPORTANT BE HELPFUL

ALWAYS READY TO LEARN NEW THINGS BE A HARDWORKING STUDENT

BE PATIENT ALWAYS CONFIDENT


39/42


Page | 39



40/42


Page | 40


y Geometry is the study of angles and triangles, perimeter, area and volume. It differs

from algebra in that one develops a logical structure where mathematical

relationships are proved and applied.

y An arithmetic progression (AP) or arithmetic sequence is

a sequence of numbers such that the difference of any two successive members of

the sequence is a constant

y A geometric progression, also known as a geometric sequence, is

a sequence of numbers where each term after the first is found by multiplying the

previous one by a fixed non-zero number called the common ratio

y Differentiation is essentially the process of finding an equation which will give you

the gradient (slope, "rise over run", etc.) at any point along the curve. Say you have

y = x^2. The equation y' = 2x will give you the gradient of y at any point along that

curve.


41/42


Page | 41



42/42


y http://www.pelajaranperak.gov.my/v2/

y http://www.facebook.com/pages/I-3-AddMath/356876395604

y http://www.facebook.com/pages/We-Can-Do-Maths/122938361050773

y http://www.facebook.com/Smkdbcmyusuf

y http://www.scribd.com/

y http://www.wikipedia.org/

y http://www.google.ca/imghp?hl=en&tab=wi

y http://www.dynamicgeometry.com/

Documents

Answers for Additional Mathematics Project Work 2 2011