Add Math Folio 2011- Corrected

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Additional Mathematics Project Work 2 2011

TABLE OF CONTENT

NO. Question Page1. Part I2. Part II

~ Question 1~ Question 2 (a)~ Question 2 (b)~ Question 2 (c)~ Question 3 (a)~ Question 3 (b)~ Question 3 (c)

3. Part III4. Further Explotion6. Reflection




ACKNOWLEDGEMENT

First of all, I would like to say thanks for giving me the chance for

doing this project work.

I would like to thank my Additional Mathematics teacher, Madam

Teoh Hai Cheu for guiding me and my friends throughout this project. We

had some difficulties in doing this project, but she taught us patiently until

we knew what to do. She tried very hard to teach us until we understand

what we supposed to do with the 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. They also supported me and encouraged

me to complete this project so that I will not procrastinate in doing it.

Last but not least, I would like to thank my friends who where doing

this project together with me and sharing our ideas. They were helpful and

cooperative when we teamed up and discussed our ideas together. So we

could complete this project work in the time given.




OBJECTIVE

The aims of carrying out this project work :

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

problems

to improve thinking skills

to promote effective mathematical communication

to develop mathematical knowledge through problem solving in a way

that increases students¶ interest and confidence

to use the language of mathematics to express mathematical ideas

precisely

to provide learning environment that stimulates and enhanceseffective learning

to develop positive attitude towards mathematics




[ PART I ]

INTRODUCTION

Cakes come in a variety of forms and flavours and are among favourite

desserts served during special occasions such as birthday parties, Hari

Raya, weddings and others. Cakes are treasured not only because of their

wonderful taste but also in the art of cake baking and cake decorating

Baking a cake offers a tasty way to practice mathematics skills, such

as fractions and ratios, in a real-world context. Many steps of baking a cake,

such as counting ingredients and setting the oven timer, provide basic

mathematics practice for young children. Older children and teenagers can

use more sophisticated mathematic to solve baking dilemmas, such as how

to make a cake recipe larger or smaller or how to determine what size

slices you should cut. Practicing mathematics while baking not only

improves your mathematics skills, it helps you become a more flexible and

resourceful baker.




MATHEMATICS IN CAKE BAKING AND CAKE DECORATING GEOMETRYTo determine suitable dimensions for the cake, to assist in designing anddecorating cakes that comes in many attractive shapes and designs, to estimatevolume of cake 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 theappropriate size and shape of pan to achieve the desired result. If the pan is toobig, the cake becomes too short. If the pan is too small, the cake becomes tootall.This leads into the next situation.The ratio of the surface area to the volumedetermines how much crust a baked good will have.The more surface area thereis, 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 topenetrate to the center. To avoid having a rock-hard outside in this case, thebaker 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 whereunmixed ingredients would accumulate, and we would have a hard time scrapingthem into the batter

CALCULUS (DIFFERENTIATION)To determine minimum or maximum amount of ingredients for cake-baking, toestimate min. or max. amount of cream needed for decorating, 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 estimatetotal amount of cream for decoration.For example when we make a cake withmany layers, we must fix the difference of diameter of the two layers. So we cansay that it used arithmetic progression. When the diameter of the first layer of thecake 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 thedifference of the diameter is constant that is2. When the diameter decreases, theweight also decreases. That is the way how the cake is balance to prevent it fromsmooch. 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 toupper layer. When we cut the cake, we can use fraction to devide the cakeaccording to the total people that will eat the cake.




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 38000cm3, and the height of thecake is to be 7.0 cm,calculate the diameter of the baking tray to be used tofit the 5 kg cake ordered by your school . [Use = 3.142]

Answer:Volume of 5kg cake = Base area of cake x Height of cake3800 x 5 = (3.142)( )² x 7

(3.142) = ( )²

863.872 = ( )²

= 29.392

d = 58.784 cm




2)The cake will be baked in an oven with inner dimensions of 80.0 cm inlength,60.0 cm width and 45.0 cm in height.(a) If the volume of the cake remains the same, explore by using differentvalues of heights,h cm, and the corresponding values of diameters of thebaking tray to be used,d cm. Tabulate your answer.

Answer: Dimension of oven

height=45cm

width= 60cmlength= 80cm

First, from the formulator d in terms of h by using the above formula for volume of cake, V=19000,that is19000 = (3.142)( )²h

=²

= d²

d =

Height,h(cm) Diameter,d(cm)1.0 155.532.0 109.983.0 89.794.0 77.765.0 69.556.0 63.497.0 58.788.0 54.99




Table 1

(b) Based on the values in your table,

(i) State the range of heights that is NOT suitable for the cake and explainyour answer.

Answer:

h < 7cm is NOT suitable, because the resulting diameter produced is toolarge 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 for the cake. Givereasons for your answer.

Answer:

The most suitable dimensions (h and d) for the cake is h = 8cm, d =54.99cm, because it can fit into the oven, and the size is suitable for easyhandling

(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.

Answer:

19000 = (3.142)( )²h

9.0 51.8410.0 49.18




=

= d²

d =

d =

log d =

log d = log h + log 155.53

Table of 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




Graph of log d against log h




(C)(ii)

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

Answer:

h when d=10.5cm

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

(b) If Best Bakery used a 42cm diameter round cake tray, use your graph toestimate the height 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 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 cakeusing the dimensions that you have suggested in 2(b)(ii)

Answer:

D=56.99

h=9cm

H=9 cm, D= 56.99 cmVolume of cream used

= volume of cake with cream ± volume of actual cake

= H - h

= 3.142 x 9 ± 3.142 x x 8

= 22960.75 ± 19002.19

=3958.56




3(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 freshcream to be used on each of the cakes.

Answer:

1 ± Rectangle-shaped base

heigh t

widthlength

19000 = base area x height

base area =

Let height= 8cm, Base area = = 2375

By trial and improvement , 50 x 47.5=2375=base area = l x w

length=50,width=47.5,height=8

Therefore, volume of cream=VC+CR ±VC =LWH ± lwh

= (52)(49.5)(9) ± (50)(47.5)(8)= 23.166- 19000= 4166




2 - Regular hexagon-shaped base

Width

Base x height = volume

Base area x 8cm = 3800 x 5

6( x l x w) x 8 =19000

x l x w =

x l x w = 395.833

Let w = 25

l x w = 791.67

l =

Length =31.667cm, width=25cm,height=8cm

Amount of cream used= volume of cake and cream ± volume of cake

= 6 [ x 26 x 33.667 ] x 9 ± 6( x 25 x 31.667 ) x 8= 23,634.234 - 19000= 4634.234




3- Pentagon-Shaped Base

width

19000 = base area x height

Let height = 8cm, base area = = 2375cm

Base area= 2375= area of 5 similar isosceles triangles in a pentagon

Therefore:

2375 = 5( l w)

950 = length x width

By trial and improvement, 950 = 50 x 19 (length=50cm , width = 19cm,height= 8cm)

Therefore, amount of cream used

= VC+CR ± V CAKE

=5( ) x 9 - 5( )x 8

=23400 ± 19000

=4400




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

No Shape Amount Of CreamUsed )(i) Rectangular shaped base 4166.00(ii) Regular hexagon-shaped base 5543.24(iii) Pentagon shaped base 4400.00(iiii) Round-shaped base 3958.56

Answer:

Round-shaped base requires the least amount of fresh cream tobe used.




Part III

Find the dimension of a 5kg round cake that requires the minimum amountof fresh cream to decorate. Use at least two different methods includingCalculus.

State whether you would choose to bake a cake of such dimensions. Givereasons for your answers.

Answer:

Conjunction

The best dimensions of the cylindrical cake is that the base dimeter is twicethe height as the surface area is the least.

R = r +1

H = h+1

V = 19000 = h = M

Vcream used = - = - = = = N

Subst M into N ,V = (2r+1)+ ]

V = [ V = +




Method 1: Calculus(Differentiation)

V = +

=38000 + +

= - - + 2

For max/min value, = 0

- = -2

2 (r + 1) =

2 (r + 1) =

2

2

2

2

r = -1 OR

(rejected)




When r = 18.22cm , d = 18.22 = 36.44cm

= 18.22cm

When r = 18.22 , > 0

Vmin

Thus, V min when d = 36.44cm

h = 18.22cm




Method 2: Table

r V

15

16

17

18

19

20

V is the minimum when r is between 18 and 19

r V

18.1

18.2

18.3

V is minimum when r is between 18.2 and 18.3




r V

18.21

18.22

18.23

From table,V is minimum when r = 18.22 d=18.22 x 2=36.44cm

h =

= 18.22 cm

Answer:

h = 18.22 cm, d=36.44cm when V is minimum




Method 3: Graph

V =²

²

From graph,

V is minimum when r = 18.22,

d = 2(18.22) = 36.44

h =²

= 18.22

33 00

33 20

334 0

336 0

338 0

34 00

34 20

344 0

15 1 6 17 1 8 19 20

V

V




FURTHER EXPLORATION

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

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

(a)Find the volume of the first, second, third and the fourth cake. Bycomparing all these values, determine whether the volumes of the cakesfrom a number pattern? Explain and elaborate on the number patterns.

Answer:

Height, h of each cake= 6cm

Radius of largest cake= 31cm

Radius of 2 nd cake = 10% smaller than 1 st cake = 27.9cm

Radius of 3rd

cake = 10% smaller than 2nd

cake = 25.11cm31, 27.9, 25.11, 22.599,«

a = 31, r =

V = (3.142) h




Radius of 1 st cake=31cm, volume of 1 st cake, = (3.142) (6)

=18116.772

Radius of 2 nd cake=27.90cm, volume of 2 nd cake , =(3.142) (6)

= 14674.585

Radius of 3 rd cake=25.11cm, volume of 3 rd cake , =(3.142) (6)=11886.414

Radius of 4 th cake=22.599cm, volume of 4 th cake = (3.142) (6)

= 9627.995

The volumes form a number pattern:

18116.772, 14674.585, 11886.414, 9627.995,«

(it is a geometric progression with first term a=18116.772 and commonratio, r = = = « = 0.81)

(b) If the total mass of all the cakes should not exceed 15kg, calculate the

maximum number of cakes that the bakery needs to bake. Verify your answer using other methods.

Answer:

=

= 15kg=15 38000 = 57000, a = 18116.772 and r = 0.81

57000 =

1- = 0.59779

0.40221 =

0.40221 = n




n =

n = 4.322

Therefore, n max = 4

Verifying the answer:

Use =

When n = 5

=

= 62104.443 > 57000 ( > 57000, n=5 is not suitable)

When n=4

=

= 54305.767 < 57000 ( < 57000, n = 4 is suitable)




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

By doing this project, I had learnt how to draw a graph and constructtables by using Microsoft Word. I had learnt how to applied mymathematical knowledge into practice like using differentiation and

progression to calculate the volume of cream used. I also learned to searchfor more information from the internet.

From this project, I had learnt some moral values. I learned that teamwork is important. Further more, I also learned to be patient always when Imeet with difficult problems, be helpful to my friends and have goodmanagement of time. Besides I learned to be more confident than before.

Actually I used to hate Additional Mathematics before because it alwaysmakes me wonder why this subject is so difficult. But now,when I learnedthat we can applied the mathematical knowledge on our daily activitiessuch as making a cake. It is so useful that and I start to love it much after conducting this project.

Team work is important

Be patient

Good management of time

Be helpful

Be a hardworking person

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Add Math Folio 2011- Corrected