addmaths project ll

8/11/2019 AddMaths Project Ll

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ADDITIONAL MATHEMATICS

PROJECT WORK 2014

TASK 2: POPCORN ANYONE?

NAME : TAN WAI SHENG

I/C NUMBER : 971005-14-5201FORM : 5 AMANAH


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CONTENTS Page

1. Acknowledgement 12. Objectives 2

3.

History 3 - 44. Moral Values 55. Section A 6 86. Section B 9 - 147. Conclusion 15


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ACKNOWLEDGEMENT

First of all, I would like to express my special thanks of gratitude to myadditional mathematics teacher, Puan Siew Yiet Hong who gave me the

opportunity to do this project and help me a lot throughout finishing thisproject. 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 providingeverything, such as money to buy anything that are related to thisproject and their advices, which is the most needed to do this project. Iam grateful for their constant support and help. Not forgotten to myfriends who have contributed lots of idea in finding the topic that wouldbe interesting to do and gave their comments on my research. I reallyappreciate their kindness and help. Besides that, I want to thanks to the

respondents for helping and spending their time to answer my questionsfor this project. Without respondents, I might not be able to completethis project because their co-operation in answering the questions, Ihave the conclusion for this project. Last but not least, I would liketo express my thankfulness to those who are involved either directly orindirectly in completing this project. Thank you for all the co-operationgiven.


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OBJECTIVES

Apply and adapt a variety of problem-solving strategies ti solveroutine and non-routine problems.

Acquire effective mathematical communication through oral andwriting, and to use the language of mathematics to expressmathematical ideas correctly and precisely.

Increase interest and confidence as well as enhance acquisitionof mathematical knowledge and skills that are useful for career andfuture undertakings.

Realize that mathematics is an important and powerful tool in solvingreal-life problems and hence develop positive attitude towardsmathematics.

Train students not only to be independent learners but alsocollaborate, to cooperate, and to share knowledge in an engaging andhealthy 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.


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HISTORY OF DIFFERENTIATION

The concept of a derivative in the sense of atangent line is a very old one,familiar toGreek geometers such asEuclid (c. 300 BC),Archimedes (c.

287212 BC) andApollonius of Perga (c. 262190 BC).Archimedes alsointroduced the use ofinfinitesimals,although these were primarily usedto study areas and volumes rather than derivatives and tangents;seeArchimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found inIndianmathematics,perhaps as early as 500 AD, when the astronomer andmathematicianAryabhata (476550) used infinitesimals to studythemotion of the moon.The use of infinitesimals to compute rates ofchange was developed significantly byBhskara II(11141185); indeed, ithas been argued that many of the key notions of differential calculus canbe found in his work, such as "Rolle's theorem". ThePersianmathematician,Sharaf al-Dn al-Ts(11351213), was the first todiscover thederivative ofcubic polynomials,an important result indifferential calculus; his Treatise on Equationsdeveloped conceptsrelated to differential calculus, such as the derivativefunction andthemaxima and minima of curves, in order to solvecubic equations whichmay not have positive solutions.

The modern development of calculus is usually credited toIsaacNewton (16431727) andGottfried Leibniz (16461716), who providedindependent and unified approaches to differentiation and derivatives.The key insight, however, that earned them this credit, wasthefundamental theorem of calculus relating differentiation andintegration: this rendered obsolete most previous methods for computingareas and volumes, which had not been significantly extended since thetime ofIbn al-Haytham (Alhazen). For their ideas on derivatives, bothNewton and Leibniz built on significant earlier work by mathematicians

such asIsaac Barrow(16301677),Ren Descartes (15961650),Christiaan Huygens (16291695),Blaise Pascal (16231662)andJohn Wallis (16161703). Isaac Barrow is generally given credit forthe early development of the derivative. Nevertheless, Newton andLeibniz remain key figures in the history of differentiation, not leastbecause Newton was the first to apply differentiation totheoreticalphysics,while Leibniz systematically developed much of the notation stillused today.
http://en.wikipedia.org/wiki/Tangent_linehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Archimedes%27_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Rolle%27s_theoremhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Rolle%27s_theoremhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Archimedes%27_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Tangent_line


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HISTORY OF POPCORN

Popcorn was first discovered thousands of years ago byNative 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 asearly as 4700 BC was found in Peru. Some Popcorn has been found in early1900s to be a purple colour. The English who came to America in the 16thand 17th centuries learned about popcorn from the Native Americans.During theGreat Depression, popcorn was comparatively cheap at 510 cents a bag and became popular. Thus, while other businesses failed,the popcorn business thrived and became a source of income for manystruggling farmers. DuringWorld War II, sugarrations diminishedcandyproduction, causing Americans to eat three times as much popcorn than

they had before. At least six localities (all in theMidwestern UnitedStates) claim to be the "Popcorn Capital of the World":Ridgway, Illinois;Valparaiso, Indiana;Van Buren, Indiana; Schaller,Iowa;Marion, Ohio; andNorth Loup, Nebraska. According to theUSDA, most of thecorn used forpopcorn production is specifically planted for this purpose; most is growninNebraska andIndiana, with increasing area inTexas. As the result ofanelementary school project, popcorn became the official statesnack food ofIllinois, U.S.A.
http://en.wikipedia.org/wiki/Indigenous_peoples_of_the_Americashttp://en.wikipedia.org/wiki/Great_Depressionhttp://en.wikipedia.org/wiki/World_War_IIhttp://en.wikipedia.org/wiki/Rationing#United_Stateshttp://en.wikipedia.org/wiki/Candyhttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Ridgway,_Illinoishttp://en.wikipedia.org/wiki/Valparaiso,_Indianahttp://en.wikipedia.org/wiki/Van_Buren,_Indianahttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Marion,_Ohiohttp://en.wikipedia.org/wiki/North_Loup,_Nebraskahttp://en.wikipedia.org/wiki/United_States_Department_of_Agriculturehttp://en.wikipedia.org/wiki/Cornhttp://en.wikipedia.org/wiki/Nebraskahttp://en.wikipedia.org/wiki/Indianahttp://en.wikipedia.org/wiki/Texashttp://en.wikipedia.org/wiki/Elementary_schoolhttp://en.wikipedia.org/wiki/Illinoishttp://en.wikipedia.org/wiki/Illinoishttp://en.wikipedia.org/wiki/Elementary_schoolhttp://en.wikipedia.org/wiki/Texashttp://en.wikipedia.org/wiki/Indianahttp://en.wikipedia.org/wiki/Nebraskahttp://en.wikipedia.org/wiki/Cornhttp://en.wikipedia.org/wiki/United_States_Department_of_Agriculturehttp://en.wikipedia.org/wiki/North_Loup,_Nebraskahttp://en.wikipedia.org/wiki/Marion,_Ohiohttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Van_Buren,_Indianahttp://en.wikipedia.org/wiki/Valparaiso,_Indianahttp://en.wikipedia.org/wiki/Ridgway,_Illinoishttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Candyhttp://en.wikipedia.org/wiki/Rationing#United_Stateshttp://en.wikipedia.org/wiki/World_War_IIhttp://en.wikipedia.org/wiki/Great_Depressionhttp://en.wikipedia.org/wiki/Indigenous_peoples_of_the_Americas


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MORAL VALUES

From the project, I learned to be independent to finish the wholeproject by myself. I also learn to time management to arrange my time

properly to accomplish this project in a shorter time.Secondly, I learnt to be responsibility. I hand in my project on time

and ask my Additional Mathematics teacher to check my project.Thirdly, I learn to help my classmates. I always prepared to help

them in the Additional Mathematics project, When we faced difficulties,we helped each other to solve the problem. I also take note during theclass when teaching was helping us to complete the project.

Lastly, I had learnt not to give up easily. I have faced manyproblems when doing this project but I try my best to solve the problems

without giving up.


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SECTION APart 1

Since the circumference of cylinder A is d = 8.5 (d is the diameter),d = 2.706 andradius = d = 1.353

Since the circumference of cylinder B is d = 11 (d is the diameter),d = 3.501 andradius = d = 1.751Therefore,

DIMENSION CYLINDER A CYLINDER B

HEIGHT 11 8.5

DIAMETER 2.706 3.501

RADIUS 1.353 1.751

Part 2The two cylinders A and B will not hold the same amount of popcorn.Cylinder B will hold more popcorn when compared to cylinder A.Since volume = r2h (r is the radius, and h is the height),r causes the volume to increase at a higher rate compared to h. This is

because r is raised to the power of two but h is not.Therefore, the volume of cylinder B is greater than that of cylinder A.

POPCORN

ANYONE?


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Part 3Cylinder B is not full.

Part 4

Yes, the prediction was correct. This is because the amount of popcornrequired to fill up cylinder A is not sufficient to fill up cylinder B.The volume of cylinder B is greater than that of cylinder A.If the prediction was incorrect, it may be due to the arrangement of thepopcorns which leave empty spaces between the popcorns and betweenthe container. It may also be due to the shoddy taping at the joints.

Part 5The formula for finding the volume of a cylinder is volume = rh.

Volume of cylinder A = (1.353)2(11) = 63.26.Volume of cylinder B = (1.751)2(8.5) = 81.87.The cylinders do not hold the same amount because the volume ofcylinder B is greater than that of cylinder A. This is because the radius israised to the power of two but the height is not. Therefore, the radiuscauses the volume to increases at a higher rate compared to the height.

Part 6

The radius impacts the volume more. This is because the radius causesthe volume to increase at a higher rate compared to the height.Therefore, the increased radius had a larger impact on the volume of thecylinder.


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SECTION B

Since 300 cm2of thin sheet material is given, the total surface area ofthe container is 300 cm2.

A cube:

Since the total surface area, 5s2= 300 cm2s = 60 cm2s = 7.746 cm

Therefore, the volume of the cube, V = (7.746cm)3= 464.764 cm3


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A cuboid with a square base:

Since the total surface area, 4sh + s2= 300 cm24sh = 300 - s2

h =

Volume, V = s2h

V = s2(

)

V = 75s -

When the volume is maximum,

= 0

75 -

= 0

= 75s2= 100s = 10 cm

Therefore, the volume of the cuboid, V = (10 cm)2(5 cm)= 500 cm3


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A circular prism (cylinder):

Since the total surface area, 2rh + r2= 300 cm22rh = 300 - r2

h =

Volume, V = r2h

V = r2(

)

V = 150r -

When the volume is maximum,

= 0

150 -

r2

= 0

r2= 150

r2= 100r = 5.642 cm

Therefore, the volume of the cylinder, V = (5.642 cm)2(5.642 cm)= 564.220 cm3


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A hemisphere:

Since the total surface area, 2r2= 300r2= 150

r = 6.910 cm

Therefore, the volume of the hemisphere, V =

r3

=

(6.910)3

= 691.023 cm3

As a consumer, I would look for a hemispherical popcorn container toobtain the maximum amount of popcorn with the same price. This will

increase the value of my money.

As a popcorn seller, I would look for a cubic popcorn container to sell theleast popcorn per unit. This is to obtain the maximum profit.

As a producer of popcorn container, I would choose to producehemispherical popcorn containers because it requires the least joints. Iwould save a large amount of cost because it does not require any bondingagent in the production.


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CONCLUSION

From this project, I have learned to determine the volume of containersof different shapes. And I have also learned that in a cylinder container,

the radius of the cylinder container increases the volume more than theheight of the cylinder container. This project taught me to economisewhen selecting a product. In this case, if we wanted the most popcorn, gofor the cylinder container. However, from this project, we also learn thatwe must be wise in handling manufacture of these containers to avoidwastage and harm towards environment. We must also consider the shapeof the container to be manufactured as it affects the cost formanufacturing it. Hence, this shows that we, as humans are gifted withminds to think and plan ahead for a better future.

addmaths project ll

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