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Part 2

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 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)(

)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 in length, 60.0cmin width and 45.0 cm in height.

a)If the volume of cake remains the same, explore by using different values of heights,hcm, and the corresponding values of diameters of the baking tray to beused,d

cm. Tabulate the answers.



Answer:

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

19000 = (3.142)(d/2)²h

=

= d²

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 58.788.0 54.99

9.0 51.84

10.0 49.18

(b)Based on the values in my table,

(i)state the range of heights that is NOTsuitable for the cakes and explain the answers.

Answer:h< 7cm is NOT suitable, because the resulting diameter produced is too large to

f it into the oven. Fur ther more, the cake would be too shor t and too wide, making it less attractive.



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

Answer:

h = 8cm, d = 54.99cm, because it can f it into the oven, and the size is suitable f or easy handling.

(c)

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

Answer:

19000 = (3.142)(

)²h

19000/(3.142)h =

= d²

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



(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 yourgraph to estimatethe 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. Thethicknessof 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 = 54.99cm Amount 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)(

)² x 1

= 2375 cm³

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 cm³

Therefore, amount of fresh cream = 2375 + 1382.23 = 3757.23 cm³

(b)Suggestthreeother shapes for cake, that will have the same height andvolume asthose suggested in 2(b)(ii). Estimate the amount of fresh cream tobe used on each 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 cm³



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 = ¥(95² + 25²)= 98.23Therefore, amount of cream= Area of rectangular front side surface(Height of cream) + 2(Area of slant rectangular left/right side surface)(Height of cream) + Vol. of top surface= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm³



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 width

By 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 cm³

(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 cm³ of cream to be used.



Par t 3

Find the dimension of a 5 kg round cake that requires the minimum amount of fresh

cream todecorate. 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.

Answer:

Method 1: Diff erentiation 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)r²h (1)V = (3.142)r² + 2(3.142)rh (2)

From (1): h =

(3)

Sub. (3) into (2):

V = (3.142)r² + 2(3.142)r(

)

V = (3.142)r² + (

)

V = (3.142)r² + 38000r -1

(

) = 2(3.142)r ± (

)

0 = 2(3.142)r ± (

) -->> minimum value, therefore

= 0

= 2(3.142)r

= r³

6047.104 = r³r = 18.22

Sub. r = 18.22 into (3):

h =

h = 18.22 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 creamis the main equation used as the quadratic function.Let f(r) = volume of cream, r = radius of round cake:19000 = (3.142)r²h (1)f(r) = (3.142)r² + 2(3.142)hr (2)From (2):f(r) = (3.142)(r² + 2hr) -->> factorize (3.142)

= (3.142)[ (r +

)² ± (

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

= (3.142)[ (r + h)² ± h² ]= (3.142)(r + h)² ± (3.142)h²(a = (3.142) (positive indicates min. value), min. value = f(r) = ±(3.142)h², correspondingvalue of x = r = --h)

Sub. r = --h into (1):19000 = (3.142)(--h)²hh³ = 6047.104h = 18.22

Sub. h = 18.22 into (1):19000 = (3.142)r²(18.22)r² = 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 are not suitable (the height is too high) and theref ore less attractive. Fur ther more,such cakes are diff icult to handle easily.



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. Theradius of the second cake is 10% less than the radius of the first cake, the radius of thethird cake is10% 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?Explain and elaborate on the number patterns.

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)r²h

Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)²(6) = 18116.772 Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585 Radius of 3rd cake = 25.11, vol. of 3rd cake = 11886.414 Radius 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 should not exceed 15 kg, calculatethe maximumnumber of cakes that 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

n =

n = 4.322

therefore, n § 4



Part 1

Cake is a form of food, typically a sweet, baked dessert. Cakes normally contain a combination

of flour, sugar, eggs, and butter or oil, with some varieties also requiring liquid (typically milk or water) and leavening agents (such as yeast or baking powder). Flavorful ingredients like fruit

purées, nuts or extracts are often added, and numerous substitutions for the primary ingredientsare possible. Cakes are often filled with fruit preserves or dessert sauces (like pastry cream), iced

with buttercream or other icings, and decorated with marzipan, piped borders or candied fruit.

Cake is often the dessert of choice for meals at ceremonial occasions, particularly weddings,anniversaries, and birthdays. There are countless cake recipes; some are bread-like, some rich

and elaborate and many are centuries old. Cake making is no longer a complicated procedure;while at one time considerable labor went into cake making (particularly the whisking of egg

foams), baking equipment and directions have been simplified that even the most amateur cook

may bake a cake.

Cake baking

y If your oven temperature is questionable, invest in an oven thermometer. Some ovens can be off by as much as 75°.

y Before mixing the batter, prepare the pans, turn the oven on, and make sure the rack is inthe center.

y Shiny pans reflect the heat, and are your best choice for cake baking.y

R educe thSubstitute 8-inch square pans for round if you want, or use 2 to 3 8 X 4-inchloaf pans. The baking time will be less, so begin checking about 15 minutes before thetime suggested.e oven temperature by 25° when using glass pans.

y Have all ingredients at room temperature for best results.y Grease pans with about 1 tablespoon of fat per layer pan.y Use cocoa (or carob powder) instead of flour for dusting a greased pan when making a

chocolate cake.

Cake decorating

I¶m here today to talk to you about a few basic tools for starting out in the world of cake

decorating. We hope you find some great cake decorating ideas. Some of them you¶re going to be very familiar with. Hopefully you have them in your own home and kitchen. Others might be

new to you, so we¶ll discuss them with you as we go along.



Obviously your going to need an apron and icing. Decorating cakes is a very messy job so I always keep an apron around. I also find having towels is very handy. Whether I¶m washing or

cleaning things and keeping things dry.

A pair of scissors, we¶ll discuss a few uses for them. Obviously your going to need a frosting

spatula. Something to scrape out the bowls when your working, when you¶re baking a cake youwant to scrape out that batter before you put it into your pans. You need to know the how much

icing or cream or whatever you're using to cover the surface of the cake. And you decide how

thick it should be. From them you multiply the thickness with the surface area of you cake to getyour icing/cream/sprinkles's volume. Then you prepare it and start decorating. As for decorating,

you decide whether you want to make your design symmetrical or whatsoever, Eg. The positionsof chocolate pieces, how many to put, how far apart... etc. That requires maths. Eg, you want to

put 6 pieces of choc on the cake. 360° divide by 6. Then use trigonometry to get how far the position chocs are.

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