NAME: DARWISH AKMAL BIN KAMAL ARIS

CLASS: 5 MEKANIKAL 1

IC NUMBER: 970826-08-6179

TEACHER’S NAME: PN. AZURA

1

ADDITIONALMATHEMATICS

POPCORN PROJECT

CONTENTS

NO

TOPICS PAGE

1. ACKNOWLEDGEMENT 32 HISTORY OF POPCORN 43. INTRODUCTION 5-94. SECTION A 10-135. SECTION B 14-186. CONCLUSION 19

2

ACKNOWLEDGEMENT

First and foremost, I would like to thank god that finally I had succeeded in finishing this project work that was given to me.

Secondly, I would like to express my special thanks gratitude to my beloved Additional Mathematics teacher, Pn. Azura who gave me the opportunity to do this project and help me a lot throughout finishing this project. I appreciate the information and advice she have given, as well as the connections she have shared with me. Without her guide, I may not finish my project and do it properly.

Besides that, I would like to thank to my parents and my family for providing everything that I needed the most such as money to buy anything that are related to this project. I am grateful as their son for their constant support and help. Not forgotten to my friends who had given me extra information on the project work and help me when I’m in lost or problem.

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 co-operations given.

3

HISTORY OF POPCORN

Popcorn Origins

The history of popcorn is deep throughout the Americas, where corn is a staple food, but the oldest popcorn known to date was found in New Mexico. Deep in a dry cave known as the "Bat Cave" small heads of corn were discovered, as well as several individual popped kernels. This discovery was made by Herbert Dick and Earle Smith in 1948. The kernels have since been carbon dated to be approximately 5,600 years old.

Decorated funeral urns in Mexico from 300 A.D. depict a maize god with popped kernels adorning his headdress. Evidence of popcorn throughout Central and South America, particularly Peru, Guatemala, and Mexico, is rampant. Aztec Indians used popcorn not only for eating, but also decoration in clothing and other ceremonial embellishments.

Native Americans throughout North America also have a rich history documenting consumption of popcorn. In addition to the kernels found in New Mexico, a kernel approximately 1,000 years old was found in Utah in a cave that was thought to be inhabited by Pueblo Indians. French explorers that came to the new world found popcorn being made by the Iroquois Indians in the Great Lakes region as well.

As colonists began moving to North America, they adopted the popular Native American snack food. Not only was popcorn eaten as a snack, but it was also reported to have been eaten with milk and sugar like a breakfast cereal. Popcorn was also cooked by colonists with a small amount of molasses, creating a snack similar to today’s kettle corn.

4

INTRODUCTION

What is volume?

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.

Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

5

Volume Formula

Shape Volume formula

Variables

Cube a = length of any side (or edge)

Cylinder r = radius of circular face, h = height

Prism B = area of the base, h = height

Rectangular prism l = length, w = width, h = height

Triangular prism b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases

Sphere r = radius of spherewhich is the integral of the surface area of a sphere

Pyramid B = area of the base, h = height of pyramid

Square pyramid s = side length of base, h = height

Rectangular pyramid

l = length, w = width, h = height

Cone r = radius of circle at base, h = distance from base to tip or height

6

CylinderA cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler") is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

The open cylinder is topologically equivalent to both the open annulus and the punctured plane.

Volume of Cylinder in diagram 1 =

Diagram 1

There are a lot of things are a lot of things around us related to cylinder or parts of a cylinder. Before I further my task, let us have a look how radius related the cylinder.

7

What is radius?

In classical geometry, the radius of a circle or sphere is the length of a line segment from its center to its perimeter. The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel. The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses. The typical abbreviation and mathematic variable name for "radius" is r. By extension, the diameter d is defined as twice the radius:

If the object does not have an obvious center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.

For regular polygons, the radius is the same as its circumradius. The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.

The radius of the circle with perimeter (circumference) C is

8

How to calculate volume of cylinder?

Having a right circular cylinder with a height h units and a base of radius r units with the coordinate axes chosen so that the origin is at the center of one base and the height is measured along the positive x-axis. A plane section at a distance of x units from the origin has an area of A(x) square units where

or

An element of volume, is a right cylinder of base area Awi square units and a thickness of Δix units. Thus if V cubic units is the volume of the right circular cylinder, by Riemann sums,

Using cylindrical coordinates, the volume can be calculated by integration over

9

Section A

QUESTION 1

For this activity, I will be comparing the sheet of paper. I will be determining which dimension can hold more popcorn. To do this, you will have to find a pattern for the dimensions for containers.

Materials: 8.5 × 11 in. white paper, 8.5 × 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 the data below and on the cylinder. Label it Cylinder A.

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

10

11in. 11in.

11in. 11 in.

8.5in. 8.5in.

8.5in. 8.5 in.

DIMENSION CYLINDER A CYLINDER BHEIGHT 11.0 8.5

DIAMETER 2.6 3.4RADIUS 1.3 1.7

2.

a) Do you think the two cylinders will hold the same amount?

- The two cylinders will hold the different amount.

b) Do you think one will hold more than other? Which one? Why?

- Cylinder B will hold more than the 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 effect of different in radius.

3. Place Cylinder B on the paper plate with Cylinder A inside it. Use your cup to pour popcorn into Cylinder A until it is full. Carefully, lift up Cylinder A so that the popcorn falls into Cylinder B. Describe what happened. Is Cylinder full, not full, or overflowing?

- Cylinder B is not full. There is still room in the cylinder for more popcorn.

As you share your popcorn snack, answer the question below.

4. a) Was your prediction correct? How do you know?

- Yes, the prediction is correct. Based on formula, radius, r has more effect than height, h since radius, r is squared. Thus, total volume of Cylinder B is bigger than Cylinder A

b) If your prediction was incorrect, describe what actually happened.

- Cylinder B has a greater volume than Cylinder A.

11

5. a) State the formula for finding the volume of a cylinder.

- V =

b) Calculate the volume of Cylinder A?

r = 1.3 in

h = 11.0 in

V = π(1.3)2(11.0)

= 58.402 in.3

c) Calculate the volume of cylinder B?

r = 1.7 in

h = 8.5 in

V = π(1.7)2(8.5)

=77.173 in.3

d) Explain why the cylinders do or do not hold the same amount. Use the formula for the volume of a cylinder to guide your explanation.

- The cylinders have different radius and heights, so the volumes are different. Based on formula, radius, r has more effect than height, h since radius, r is squared

12

6. Which 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.

CYLINDER RADIUS HEIGHT VOLUMEORIGINAL 3 10 282.7 in3

INCREASED RADIUS

4 10 502.7 in3

INCREASED HEIGHT

3 11 311.0 in3

Which increased the 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.

13

SECTION B

If you were buying popcorn at the movie theatre and wanted the most popcorn, what type of container would you look for?

You are given 300 cm2 of thin sheet material. Explain the details.

1. Cylinder Container – opened top

14

2. Cube Container – opened top

Surface Area = ι2 +ι2 = 300

ι2 = 60

ι = 7.75 cm3

Volume = ι3

= 465.48

3. Cuboid Container – opened top

h=300−2(7.07)2

4 (7.07 )

h=7.07 cm

Volume = 2l²h = 2(7.07)²(7.07)

15

= 706.79cm³

4. Cuboid Container – opened top

Volume = 500 cm3

16

5. Hexagon Container – opened top

17

6. Cone Container – opened top

Volume = 606.197

18

CONCLUSIONContainer Height Radius Length Width VolumeCylinder 5.64 5.64 - - 563.69

Cube 7.75 - 7.75 7.75 465.48Cuboid 1 7.07 - 7.07 14.14 706.79Cuboid 2 5 - 10 10 500.00Hexagon 9.49 - 4.39(side) - 475.17

Cone 10.51 7.42 - - 606.197

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 1 can be filled in with the most amount popcorn. It followed by cone, cuboid 2, and hexagon. These means that cube is the container that can be filled with the least amount of popcorn. Randomly, surveying at the movie theater, no hexagon or cube shapes can be found. Therefore, in this case, the cuboid 1 was the most preferable container that can have the most popcorns.

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

- If I was the popcorn seller, I would go for cube container because the least popcorns will be in. So, I can obtain the most profit for my sale in the future. Furthermore, it is cute and simple shape.

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

- I would choose cylinder shape since it is the easiest production and it takes less effort and no time consuming to produce.

19