ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICSAPPLICATION OF MATHEMATICS IN POPCORN PACKAGING

ACKNOWLEDGEMENTFirst of all, I would like to express my special thanks of gratitude to my additional mathematics teacher, Puan Rubiah who gave me the opportunity to do this project and help me a lot throughout finishing this project. Without her guide, I may not finish my project and do it properly. Secondly, I would like to thanks my parents and my family for providing everything, such as money to buy anything that are related to this project and their advises, which is the most needed to do this project. I am grateful for their constant support and help. Not forgotten Izzah , my friends who have contributed lots of idea in finding the topic that would be interesting to do and gave their comments on my research. I really appreciate their kindness and help. Beside that, I want to thanks to the respondents for helping and spending their time to answer my questions for this project. Without respondents, I might not be able to complete this project because their co-operation in answering the questions, I have the conclusion for this project. Last but not least, I would like to express my thankfulness to those who are involved either directly or indirectly in completing this project. Thank you for all the co-operation given.

contents

NOTOPICSPAGE

1INTRODUCTION

2ACKNOWLEDGEMENT

3HISTORY

4OBJECTIVE

5SECTION A

6SECTION B

7CONCLUSION

INTRODUCTION TO POPCORN

Popcorn is a type of corn that expands from the kernel and puffs up when heated. Popcorn is able to pop like amaranth grain, sorghum, quinoa, and millet. When heated, pressure builds within the kernel, and a small explosion (or "pop") is the end result. Some strains of corn are now cultivated specifically as popping corns.There are various techniques for popping corn. Along with prepackaged popcorn, which is generally intended to be prepared in a microwave oven, there are small home appliances for popping corn. These methods require the use of minimally processed popping corn.A larger-scale, commercial popcorn machine was invented by Charles Cretors in the late 19th century.Unpopped popcorn is considered nonperishable and will last indefinitely if stored in ideal conditions.Depending on how it is prepared and cooked, some consider popcorn to be a health food, while others caution against it for a variety of reasons. Popcorn can also have non-food applications, ranging from holiday decorations to packaging materials.

HISTORY OF POPCORN

Popcorn was first discovered thousands of years ago by Native Americans. It is one of the oldest forms of corn: evidence of popcorn from 3600 B.C. was found in New Mexico and even earlier evidence dating to perhaps as early as 4700 BC was found in Peru. Some Popcorn has been found in early 1900s to be a purple color. The English who came to America in the 16th and 17th centuries learned about popcorn from the Native Americans.During the Great Depression, popcorn was comparatively cheap at 5 10 cents a bag and became popular. Thus, while other businesses failed, the popcorn business thrived and became asource of income for many struggling farmers. During World War II, sugar rations diminished candy production, causing Americans to eat three times as much popcorn than they had before.At least six localities (all in the Midwestern United States)claim to be the "Popcorn Capital of the World": Ridgway, Illinois; Valparaiso, Indiana; Van Buren, Indiana; Schaller, Iowa; Marion, Ohio; and North Loup, Nebraska. According to the USDA, most of the corn used for popcorn production is specifically planted for this purpose; most is grown in Nebraska and Indiana, with increasing area in Texas. As the result of an elementary school project, popcorn became the official state snack food of Illinois ,U.S.A.

OBJECTIVE

Apply and adapt a variety of problem-solving strategies to solve routine and non-routine problems.Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely.Increase interest and confidence as well as enhance acquisition of mathematical knowledge and skills that are useful for career and future undertakings.Realize that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.Train students not only to be independent learners but also collaborate, to cooperate, and to share knowledge in an engaging and healthy environment.Use technology especially the ICT appropriately and effectively.Train students to appreciate the intrinsic values of mathematics and to become more creative and innovative.Realize the importance and the beauty of mathematics.

section aQUESTION 1For this activity, you will be comparing the volume of 2 cylinders created using the same sheetof paper. You will be determining which dimension can hold more popcorn. To do this, youwill have to find a pattern for the dimensions for the containers.Materials : 8.5 x 11 in. white paper, 8.5 x 11 in. colored paper, tape, popcorn plate, cup, ruler.1. Take the white paper and roll it up along the longest side to form a baseless cylinder that Is tall and narrow. Do not overlap the sides. Tape along the edges. Measure the dimensions with a ruler and record your data below and on the cylinder. Label it Cylinder A.

2. Take the colored paper and roll it up along the shorter side to form a baseless cylinderthat is short and stout. Do not overlap the sides. Tape along the edge. Measure the heightand diameter with a ruler and record you data below and on the cylinder. Label it Cylinder B.

DIMENSIONCYLINDER ACYLINDER B

HEIGHT118.5

DIAMETER2.63.4

RADIUS1.31.7

QUESTION 2Do you think the two cylinders will hold the same amount? Do you think one will hold more than the other? Which one? Why?ANSWER:The two cylinders will hold the different amount. Cylinder B will hold more than Cylinder A. This is because the radius of Cylinder B is longer and this make the volume is bigger than Cylinder A. Although the height of Cylinder B is shorter than Cylinder A, but this does not affect much compare the affect of different in radius.

QUESTION 3Place Cylinder B on the paper plate with Cylinder A inside it. Use your cup to pour popcorninto Cylinder A until is full. Carefully, lift Cylinder A so that the popcorn falls into Cylinder B.Describe what happened. Is Cylinder B full, not full or over flowing?ANSWER:Cylinder B is not full. There is still space in the cylinder for more popcorn.

QUESTION 4a) Was your prediction correct? How do you know?b) If your prediction is incorrect, describe what actually happenedANSWER:a) Yes, the prediction is correct. It is based on the formula, volume of cylinder equals to . According to the formula, radius, r has more effect than height, h since radius, r is squared. Thus, the Cylinder B with greater radius, r have the greater volume, V than Cylinder A.b) Cylinder B has a greater volume than Cylinder A.

QUESTION 5a) State the formula for finding the volume of a cylinderb) Calculate the volume of Cylinder A.c) Calculate the volume of Cylinder Bd) Explain why the cylinders do or do not hold the same amount. Use the formula for the formula for the volume of a cylinder to guide your explanation.ANSWER:a) b) c) d) The cylinders have different radius and heights, so the volumes are different.

DATA AND OBSERVATION The cylinder with have the greater radius and diameter will have the greater volume The radius of Cylinder B is greater than Cylinder A. The volume of Cylinder B is greater than Cylinder A. So, Cylinder B holds more popcorn than Cylinder B.DIMENSIONCYLINDER ACYLINDER B

HEIGHT, inc118.5

DIAMETER, inc2.63.4

RADIUS, inc1.31.7

VOLUME, inc58.477.2

QUESTION 6Which measurement impacts the volume more: the radius or the height? Work through the example below to help you answer the question.Assume that you have a cylinder with a radius of 3 inches and a height of 10 inches. Increase the radius by 1 inch and determine the new volume. Then using the original radius, increase the height by 1 inch and determine the new volume.ANSWER:CYLINDERRADIUSHEIGHTVOLUME

ORIGINAL310282.7

INCREASED RADIUS410502.7

INCREASED HEIGHT311311

Which increased dimension had a larger impact on the volume of the cylinder? Why do you think this is true?Increasing the radius increased the volume more than increasing the height. This is because the radius is squared to find the volume, which increases its impact on the volume.

SECTION BQUESTIONIf you were buying popcorn at the movie theater and wanted the most popcorn, what type of container would you look for?Clue: You need more than one type of containers.You are given 300 cm of thin sheet material. Explain the details.ANSWER:1- CYLINDER CONTAINER OPENED TOP

(Maximum volume= = = h= = h= 5.64cm r = 5.64 cm VOLUME = 563.62 cm2- CUBE CONTAINER OPENED TOP

Surface Area = + 4 = 300cm 5 = 300 = 60 = 7.75cm Volume = = (7.75) = 465.48 cm

3- CUBOID CONTAINER OPENED TOP Assume that length is twice its width or othersSurface Area = 2 +4h = 300cm h = Volume = 2 h 2 ( Maximum volume = 2 ( = ( = 7.07 cm = ( h = = 150 - h = 7.07 cm =0 VOLUME = 2 150 = 3 = 2 = 706.79 c

4- CONE CONTAINER OPENED TOPFrom the diagram, x = r + h Surface Area = r x = 300cm r x = 300 r ( r + h) = 90000 Volume = volum = )Maximum volume = r = 7.42 h = 10.51VOLUME = = 605.95 c

conclusionCONTAINERHEIGHTRADIUSLENGTHWIDTHVOLUME

CYLINDER5.645.64 - -563.69

CUBE7.75 -7.757.75465.48

CUBOID7.07 -7.0714.14706.79

CONE10.517.42 - -605.95

Shape of containers that give the most popcorn reflect the maximum volume. From the activity earlier, I found that increasing the radius increased the volume more than increasing the height. This is because the radius is squared to find the volume, which increases its impact on the volume. From the calculations, it has been found that cuboid can be filled in with the most amount popcorn. These means that cube is the container that can be filled with the least amount of popcorn. Randomly, surveying at the movie theatre , no cube or cuboid shapes can be found. Therefore, in this case, the cuboid was the most preferable container that can have the most popcorns.

You are the popcorn seller, what type of container would you look for?If I was the popcorn seller, I will look for cube shape container. It is because the least popcorns will be in. So, I will get the most profit for my sale. Furthermore, it is cute and simple shape.

You are the producer of the containers, what type of container would you choose to have the most profit?If I was the producer of the popcorns containers, I will look for cylinder shaped container. It is because this shape is the easiest production and it takes less effort and also no time consuming to produce.

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