Add Maths Project [Complete]

Table of Contents

No. Content Page1. Introduction2. Moral Values3. History of Popcorn4. Objectives5. Questions6. Section A7. Section B8. Reflection9. Conclusion

Containers have existed in our life ever since the dawn of mankind. Humans have

used containers even before houses were built. Containers are important in our lives

because it helps us hold onto heavy items, prevent items from dropping, preservation and

also the measurement of non-solid things. Without containers, we cannot even store food

for more than 1 day. Containers can be made from different materials. We usually see

containers that are made from iron, plastic and even paper.

Containers can come in different size and shape. The most common container that

we can see is the container used to hold popcorn in cinemas. The containers used to hold

popcorn can vary from square shaped to cylinder shaped and to cone shaped. Without this

containers, we have to hold the popcorn with our bare hands. This is not only a burden, but

also unhygienic.

Mathematics is used in creating a container. Without mathematics, we cannot

calculate the capacity of a container and the number of items it can hold. For example, we

can calculate the volume of a square container using the formula for the volume of square

(base x height x width).

From this additional mathematics project, I have learned many moral values. I have

learned to be grateful to be living in this modern age where we can use mathematics to

calculate even the slightest details to solve a problem. Without this, our world wil still be in

the prehistoric age where humans live in caves and hunt for food.

I am also grateful to the scientists who invented the container. Without a container,

we would not be able to hold stuff without things falling off. We could not even hold water

if there was no container in this world. This discovery has taught me to be thankful for the

invention of the container.

Finally, I am also grateful to have a kind teacher, parents and also friends to help me

complete this project. Without them, I would not be able to complete this project in time.

With their guidance, I can finally complete this assignment.

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 a source 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.

OBJECTIVESApply and adapt a variety of problem-solving strategies ti 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.

1) White Paper Cylinder (CYLINDER A)

2) Coloured Paper Cylinder( CYLINDER B)

SECTI

1)

DIMENSION CYLINDER A CYLINDER B

HEIGHT 11 Inches 8.5 Inches

DIAMETER 2.70 Inches 3.50 Inches

RADIUS 1.35 Inches 1.75 Inches

2.i) No.

ii) I think that cylinder B will hold more than cylinder A. The volume of cylinder B is greater

than cylinder A because of the difference in radius.

3) When cylinder A is removed, the popcorn falls into cylinder B. However, the popcorn did

not fill up cylinder B.

4)a) My prediction was correct. The popcorn used for both cylinders are equal in amount.

5a)Volume of cylinder =π r2h

b)Volume of cylinder A = 227×1.352 inc hes×11inc hes = 63.01 inc hes2

c) Volume of cylinder B = 227×1.752 inc hes×8.5 inc hes

= 81.81inc hes2

d) According to the volume of the cylinder, cylinder B has a greater volume than cylinder A.

When the volume differs, the amount that can be held differs. Hence, cylinder B will hold

more popcorn than cylinder A.

6.a)Increase in Radius

Radius Height Volume

3 10282.857

1

4 10502.857

1

5 10785.714

3

6 101131.42

97 10 1540

8 102011.42

9

9 102545.71

4

10 103142.85

7

11 103802.85

7

12 104525.71

4

13 105311.42

9

b) Increase in Height

Radius Height Volume

3 10282.857

1

3 11311.142

93 12 339.428

6

3 13367.714

33 14 396

3 15424.285

7

3 16452.571

4

3 17480.857

1

3 18509.142

9

3 19537.428

6

3 20565.714

3

The increase in radius has a larger impact on the volume because radius is directly

proportional to volume.

Height,h Diameter,d Radius,r Volume,V

Cylinder A 11 2.6 1.3 58.4

Cylinder B 8.5 3.4 1.7 77.2

5

15

25

35

45

55

65

75

85

Data and Observations:

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.

SECTI

ON B

SECTION B

Question:

If you were buying popcorn at the movie theatre 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 300cm² of thin sheet material. Explain the details.

Answer:

Identify Problem:

To find a suitable shape of a popcorn container that can contain the most popcorn.

Strategy:

Compare the volume of 3 different types of popcorn containers, which is the frustum shape, cone shape and rectangular shape.

x

i) Frustum of square base with an open top container

Strategy:

1. Use the formula, area = x2+ ( x+r )(12 )(h )(4 ), where x is the length of the base, r is the

length of the open surface and h is the height of the trapezium surface, to find the

suitable length of each side of the frustum shaped container.

2. By using trial and error and the maximum surface area, which is 300cm2, to

determine the most suitable lengths of each side for the frustum shaped container.

3. By using the dimensions found, find the volume of the frustum using the formula,

volume = h3(x2+r 2+√x2 r2)

Working:

Since area = x2+ ( x+r )(12 )(h )(4 ),

By using trial and error, let x = 7, and r = 3,

Find the length of h.

300 = 72+(7+3 )( 12 ) (h )(4)

300 = 49+20h

251 = 20h

h = 12.55 cm

When h = 12.55, x = 7 and r = 3,

Volume = h3(x2+r 2+√x2 r2)

= 12.55

3(72+32+√7232)

= 274.493 cm3

Conclusion:

The volume of a frustum of pyramid square shaped base with open an top is 274.493 cm3

ii) Cone shape with open base container

Strategy:

1. Using the formula, area of cone = , where π = 3.142 , r = radius and s = slanted

height of the cone, without adding the area of the base as the cone has an open

base.

2. Using the radius, slanted height and height of the cone found, find the volume of the

container using the formula 13π r2h, where π = 3.142, r = radius and h = height.

Working:

Area = πrs

= πr (√r2+h2)

3002 = π2 r2(√r2+h2)

= π2 r4+π2 r2h2

h2 = 90000−π2 r4

π 2r2 -------- (1)

V=13π r2h

V 2=19π2 r4 --------- (2)

Sub (1) to (2)

V 2=19π2 r4 ¿ )

V 2=10000 r2−π2 r6

9

dvdr

=10000 r2−π2r6

9

When dvdr

=0,

10000−π2 r4

3=0

π 2r4

3=10000

r=7.43cm

h=10.51cm

V=13π r2h

V=13π (7.43¿¿2)(10.51)¿

V=606.03cm3

Conclusion:

Therefore, the volume of the cone shaped container with an open base is 606.03cm3

2l

l

iii) A cuboid shaped container with an open top

Strategy:

1. Use a length of 2l, width of l and height of h to find value of l with differentiation

method.

2. By using the equation 2l2 + 6hl= 300cm2, make an equation of h in terms of l.

Substitute the equation of h in terms of l into the equation V = 2l2h.

3. Then using the value of l found, find the volume of the cuboid using equation V =

2l2h.

Working:

2l2 + 6hl = 300cm2

6hl = 300cm2 - 2l2

h = 300−2 l2

6 l---- (1)

V = 2l2h ---- (2)

Substitute (1) into (2)

V = 2l2 (300−2 l2

6 l)

= 100l - 23 l3

dvdl

= 0

dvdl

= 100 - 2 l2

100 - 2 l2 = 0

2 l2 = 100

l2 = 50

h

l = 7.0711

V = 150(7.0711) – (7.0711) 3

= 707.1068cm 3

Conclusion:

Therefore, the volume of the cuboid with an open top is 707.1068cm3.

Frustum

Cone

Cuboid

274.493

606.03

707.1068

Volume of container,V (cm³)Volume of container

Answer:

If I were a consumer who is searching for a popcorn container that would offer the most value-for-money amount of popcorn, I would look for the container that is shaped like a cuboid as it has the largest volume among all of the three containers. When the cuboid shaped container is used, I would pay the same amount of money for more popcorn. This is because a larger container with a larger volume used would mean that the container would be able to hold more popcorn compared to the other containers.

Question:

Have a thought about:

i. You are the popcorn seller, what type of container would you look for?

ii. You are the producer of the containers, what type of container would you choose to have the most profit.

Answer:

i) If I were the popcorn seller, the most suitable container for me would be the frustum shaped container as it has the smallest volume out of the three containers and can hold the least amount of popcorn. When a container with a smaller volume is used, I will be able to obtain a higher profit as I will be able to sell less popcorn for the same price.

ii) If I were the producer, I would choose the frustum shaped container too because when the profit rate for the popcorn seller is high, he will be able to earn more money. Thus, his demand for the cone shaped containers that is profitable for him will increase and as a producer, I can sell more containers to him and earn more money.

REFLECTION

This project has taught me lots of things. I definitely did not expect for additional mathematics to be so useful in reality.

I guess, in reality, additional mathematics has taught me to bond with people and toughen myself up. It is a really tough subject and has caused me to ask my seniors and teachers for their guidance. It has also taught me patience, for I have to make a lot of practice in order to improve my additional mathematics knowledge.

Hence, this is a really short and original poem regarding add math or rather my add math teacher.

I am done, I am through,

I am so sick of it all,

I am tired of the equations,

And the solutions of log.

But you came and you see,

Yeah you taught it all to me,

I am glad to have a teacher (you see),

That teaches add math to me.

Thank you!!

Conclusion

After doing research, answering questions and drawing bar charts, I finally understood the importance of the usage of geometry and calculus in daily life.

Geometry is the study of angles and triangles, perimeter, area and volume. It differs from algebra under that rule where one develops a logical structure where mathematical relationships are proved and applied.

Differentiation is essentially the process of finding a gradient from an equation at any point along the curve. Say you have the equation, y=x2. The equation, y=x2, will give you the gradient of y at any point along the curve.

As a conclusion, geometry, calculus and progressions are part of our necessity. Thus, we should be thankful to the people who had contributed their support and ideals to the creation of these mathematical rules.