ADDITIONAL MATHEMATICS PROJECT WORK 2011 GROUP MEMBERS: CONTENT PAGE |No. |Content |Page(s) | |1. |Introduction | | |2. |Definition | | |3. |History | | |4. |Part I | | |5. Part II | | |6. |Part III | | |7. |Further Exploration | | |8. |Reflection | | INTRODUCTION We students taking Additional Mathematics are required to carry out a project work while we are in Form Five.
This year the Curriculum Development Division Ministry of Education has prepared four tasks for us. We are to choose and complete only ONE task based on our area of interest. This project can be done in groups or individually, but each of us are expected to submit an individually written report. I think the additional mathematics project work is a very good opportunity for us to test and enhance our thinking skills. We are given 6 weeks time frame to complete the assignment and I believe I have the luxury of time to think, think and think over the solutions to those questions.
And the point of all of this is to learn new things and explore new horizons in mathematical fields. On this project I am working on, I hope to be able to make generalization about the volume and the weight of a cake for the occasion of Teachers Day celebration. Through the use of various mathematical methods like calculus, trial-and-improve method and graph, I am certainly and fairly to be able to achhieve my goal. I would like to point out that by using correct and suitable mathematical method and calculation, I sincerely wish that my project will be able to be implemented and achieve the desired results.
Definition Pi, ? has the value of 3. 14159265. In Euclidean plane geometry, ? is defined as the ratio of a circle’s circumference to its diameter. [pic] ? =[pic] The ratio [pic] is constant, regardless of a circle’s size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, C, preserving the ratio [pic]. Alternatively ? can also be defined as the ratio of a circle’s area (A) to the area of a square whose side is equal to the radius. [pic] ? =[pic] A BRIEF HISTORY AND FACTS ABOUT CAKES
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 purees, nuts or extracts are often added, and numerous substitutions for the primary ingredients are 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. Cakes are broadly divided into several categories, based primarily on ingredients and cooking techniques. Yeast cakes are the oldest, and are very similar to yeast breads. Such cakes are often very traditional in form, and include such pastries as babka and stollen. • Cheesecakes, despite their name, aren’t really cakes at all. Cheesecakes are in fact custard pies, with a filling made mostly of some form of cheese (often cream cheese, mascarpone, ricotta or the like), and have very little to no flour added, although a flour-based crust may be used. Cheesecakes are also very old, with evidence of honey-sweetened cakes dating back to ancient Greece. Sponge cakes are thought to be the first of the non-yeast-based cakes and rely primarily on trapped air in a protein matrix (generally of beaten eggs) to provide leavening, sometimes with a bit of baking powder or other chemical leaven added as insurance. Such cakes include the Italian/Jewish pan di Spagna and the French Genoise. Highly decorated sponge cakes with lavish toppings are sometimes called gateau; the French word for cake. • Butter cakes, including the pound cake and devil’s food cake, rely on the combination of butter, eggs, and sometimes baking powder to provide both lift and a moist texture.
Type of cakes commonly found in Malaysia |Birthday cake |Span cake |Banana cake |Butter cake | |[pic] |[pic] |[pic] |[pic] | |Lemon cake |Pound cake |Short cake |Orange cake | |[pic] |[pic] |[pic] |[pic] | |Spice cake Carrot cake |Fish cake |Pancakes | |[pic] |[pic] |[pic] |[pic] | |Sponge cake |Cheese cake |Layer cake |Cup cake | |[pic] |[pic] |[pic] |[pic] |
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, birthday, celebration and etc. Cakes use mostly some form of cheese which is often cream cheese, or ricotta, and have very little to no flour component, although it sometimes appears in the form of a sweetened crust. he purpose behind cake decorating is to turn an ordinary cake into a spectacular piece of food art. Decorating a cake can be as complex or as simple as you wish.
Cakes are treasured not only because of their wonderful taste but also in the art of cake baking and cake decorating. Sponge cake is a cake based on flour (usually wheat flour), sugar, and eggs, sometimes leavened with baking powder, that derives its structure from an egg foam into which the other ingredients are folded. In addition to being eaten on its own, it lends itself to incorporation in a vast variety of recipes in which pre-made sponge cake serves as the base. The sponge cake is thought to be one of the first of the non-yeasted cakes. Part I
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 etc. Cakes are treasured not only because of their wonderful taste but also in the art of cake baking and cake decorating. Find out how mathematics is used in cake baking and cake decorating and write about your findings. Answer: We can use the geometry method to determine suitable dimensions for the cake, to assist in designing and decorating cakes that comes in many attractive shapes and designs and to estimate volume of cake to be produced
Other than that, calculus or specified as differentiation is used to determine minimum or maximum amount of ingredients for cake-baking and cream needed for decorating and to estimate the minimum or maximum size of cake produced. Besides that, progressions is used to determine total weight and the volume of multi-storey cakes with proportional dimensions, to estimate total ingredients needed for cake-baking and total amount of cream for decoration. 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. 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 your school. [Use ? = 3. 142] Answer: Volume of 5kg cake = Base area of cake x Height of cake 3800 x 5 = (3. 142)(r)? x 7 (3. 142) = r? 863. 872 = r? r = 29. 392 d = 2r = 2 x 29. 392 = 58. 784 cm 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. ) 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)(r)? h 19000/3. 142h = d2/4 24188. 145/h= d2/4 d = 155. 53/ h |Height,h (cm) |Diameter,d(cm) | |1. |155. 53 | |2. 0 |109. 98 | |3. 0 |89. 80 | |4. 0 |77. 77 | |5. |68. 56 | |6. 0 |63. 49 | |7. 0 |58. 78 | |8. 0 |54. 99 | |9. |51. 84 | |10. 0 |49. 18 | (b) Based on the values in your table, (i) State the range of heights that is NOT suitable for the cakes and explain your answers. Answer: The range h ; 7cm is not suitable to make the cake because the resulting diameter produced is too large to fit into the oven.
Furthermore, the cake would be too short and too wide, making it less attractive and hardly to handle. (ii) Suggest the dimensions that you think most suitable for the cake. Give reasons for your answer. Answer: The cake with the dimension h=7cm and d=54. 99cm. It is because the cake is fit to put into the oven, and the size is easier to handle. (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. [You may draw your graph with the aid of computer software. ] Answer: 19000 = (3. 142)x r? h 19000/3. 142h= d2/4 24188. 145/h=d2/4 d=155. 53h-1/2 log10d=-1/2 log10h + log10155. 53 19000 / (3. 142)h = |Log10 h |0 |1 |2 |3 |4 | |Log10 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. 5 cm, use your graph to determine the diameter of the round cake pan required. Answer: h = 10. 5cm, log h = 1. 21, log d = 1. 680, d = 47. 86cm (b) If Best Bakery used a 42 cm diameter round cake tray, use your graph to estimate 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 1cm (a) Estimate the amount of fresh cream required to decorate the cake using the dimensions that you have suggested in 2(b)(ii). Answer: h = 8cm, d = 54. 99cm Volume of cake before decoration = (3. 142) x (54. 99/2)2 x 8 = 19002. 18 cm3
New volume of cake after decoration = (3. 142) x (54. 99/2 +1)2 x 9 = 3. 142 x 28. 4922 x9 = 22960. 75 Therefore, amount of fresh cream= 22960. 75 – 19002. 18 = 3958. 57 cm3 (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 to be used on each of the cakes. Answer: Rectangle-shaped base (cuboid) 19002 = base area x height base area = 19002 / height length x width = 2375. 25 By trial and improvement, 2375. 25 = 50 x 47. 505 (length = 50, width = 47. 505, height = 8) Therefore, the amount of cream needed 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. 505)(1) + 2375. 25 = 3935. 33 cm Pentagon-shaped base 19002 = base area x height base area = 2375. 25 = area of 5 similar isosceles triangles in a pentagon therefore: 2375. 25 = 5(length x width) 475. 05 = length x width By trial and improvement, 475. 05 = 25 x 19 (length = 25, width = 19) Therefore, the amount of cream needed = 5(area of one rectangular side surface)(height of cream) + vol. of top surface = 5(8 x 19) + 2375. 25 = 3135. 5 cm? Square-based shape 19002 = side2 x height Side2 =2375. 25 Side = 48. 74 (width=48. 74cm, length=48. 74cm, height=8cm) Therefore, the amount of cream needed = vol of top surface + 4(area of one rectangular side surface)(height of cream) = 2375. 25 + (4 x 8 x 48. 75 x 1 ) = 3935. 25 cm3 (c) Based on the values that you have found which shape requires the least amount of fresh cream to be used? Answer: Pentagon-shaped cake, since it requires only 3135 cm? of cream to be used. Part III Find the dimension of a 5 kg round cake that requires the minimum amount of fresh cream to decorate.
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: Differentiation Use two equations for this method: the formula for volume of 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)r? h > (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 = r? r = 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 cream is the main equation used as the quadratic function. Let f(r) = volume of cream, r = radius of round cake: 19000 = (3. 142)r? h > (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. 42)[ (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)? h h? = 6047. 104 h = 18. 22 Sub. h = 18. 22 into (1): 19000 = (3. 142)r? (18. 22) r? = 331. 894 r = 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 therefore less attractive. Furthermore, such cakes are difficult to handle. FURTHER EXPLORATION
Best Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, as shown in Diagram 2. [pic] The height of each cake is 6. 0 cm and the radius of the largest cake is 31. 0 cm. The radius of the second cake is 10% less than the radius of the first cake, the radius of the third 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 comparing all these values, determine whether the volumes of the cakes form a number pattern? Explain and elaborate on the number patterns. Answer: eight, h of each cake = 6cm radius of largest cake = 31cm radius of 2nd cake = 10% smaller than 1st cake =27. 9 radius of 3rd cake = 10% smaller than 2nd cake = 25. 11 Radius of 4th cake= 10% smaller than 3rd cake =25. 599 a = 31, r = V = (3. 142)r? h Radius of 1st cake = 31, Volume of 1st cake = (3. 142)(31)? (6) = 18116. 772 Radius of 2nd cake = 27. 9, Volume of 2nd cake = 14674. 585 Radius of 3rd cake = 25. 11, Volume of 3rd cake = 11886. 414 Radius of 4th cake = 22. 599, Volume of 4th cake = 9627. 995 The progression is as following: 18116. 772, 14674. 585, 11886. 414, 9627. 95, … a = 18116. 772, common ratio, r = T2/T1 = T3 /T2 = … = 0. 81 Thus, it is a geometric progression with the first term, a = 18116. 772 and the common ratio, r= 0. 81. (b) If the total mass of all the cakes should not exceed 15 kg, calculate the maximum number of cakes that the bakery needs to bake. Verify your answer using other methods. Answer: The volume of 1kg of cake = 3800 Thus, the volume of 15kg of cake = 3800 x 15 =57000 Sn ; 57000 a (1-rn) / 1-r ;57000 18116. 772 ( 1-0. 81n) / 1. 0. 81 ; 57000 18116. 772 (1-0. 81n) ; 10830 1-0. 81n ; 0. 5978 0. 81n ; 0. 4022 log100. 81 ; log10 0. 4022 n ; 4. 322 therefore, n = 4 REFLECTION In the making of this project, I have spent countless hours doing this project. I realized that this subject is a compulsory to me. Without it, I can’t fulfill my big dreams and wishes…. I used to hate Additional Mathematics… It always makes me wonder why this subject is so difficult… I always tried to love every part of it… It always an absolute obstacle for me… Throughout day and night… I sacrificed my precious time to have fun… From now on, I will do my best on every second that I will learn Additional Mathematics.