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1. Add Math Project 2/2016
Name: Teh Ming Yang
Class: 5C
MY card number: 991216-14-6157
School: Wesley Methodist School Kuala Lumpur (International)
Teacher’s name: Madam Sim Hui Hui
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2. Content
Item Page number
Content 2
Introduction 3-4
Content 5-7
Section A 8-14
Section B 15-23
Conclusion 24
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3. Introduction
In everyday lives, we technically apply mathematics in everything.
Calculating the volume of a solid is an example of the importance of mathematics in
our daily lives. In the course of this project, the below mathematical principles
explained will be applied.
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, m³. The volume of a container is generally understood
to be the capacity of the container, for example 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.
Surface area is the area of the outside of a solid. Surface area, just like area,
uses the same SI derived unit, which is square metre, m². The surface area of a solid is
generally understood as the outer surface of an object which has contact with the
atmosphere. In other words, surface area is the exterior of a solid. Surface area of
solids can be calculated easily by using formulas. Surface area is extremely useful
when we want to know how much of a material has to be used to model a solid by
completing all sides of the solid.
A linear equation is an algebraic equation in which each term is either a
constant or the product of a constant and (the first power of) a single variable. Linear
equations can have one or more variables. Linear equations occur with great
regularity in applied mathematics. While they arise quite naturally when modelling
many phenomena, they are particularly useful since many non-linear equations may
be reduced to linear equations by assuming that quantities of interest vary to only a
small extent from some "background" state.
A quadratic equation is similar to a linear equation, only with the variables
having different powers, usually squared and also to the power of one. Quadratic
equation can either have one root (or two similar roots), two distinct roots or no roots
at all. Graphs of quadratic equation always results in a curve graph, with either a
maximum value or a minimum value, depending on the value of the variable in the
equation. In general form, a quadratic equation will be in the form of
, where and are variables.
In mathematics, differentiation is used to calculate the maximum value of an
equation. It is useful in our daily lives by allowing us to calculate and determine the
maximum volume of a solid. It can also be used to determine the maximum value of
other things which requires a quadratic equation. Second differentiation, on the other
hand, allow us to determine whether the value we obtained earlier is a maximum
value or a minimum value. This is done by differentiating the differentiated value
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4. from the previous differentiation process. This is particularly useful for us to do to
calculation to find the largest or smallest value.
The objective of doing this project is to be able to apply and adapt a variety of
problem-solving strategies to solve routine and non-routine problems. We tend to be
able to learn different formulas in mathematics but unable to apply them fully in our
daily lives when we need to. This project aims to provide an environment for us to be
able to apply the knowledge we learned in our daily lives when faced with problems.
The other objective is to increase our interest and confidence as well as enhance
acquisition of mathematical knowledge and skills that are useful for career and future
undertakings. Mathematics is all about solving problems and we learn mathematics to
solve a problem correctly. Many careers are also heavily related to mathematics and
by having the necessary mathematical skills, we will be able to excel. This project
also makes us realize that mathematics is an important and powerful tool in solving
real-life problems hence develop positive attitude towards mathematics. We usually
take mathematics for granted and usually do not realize how important it can play in
our daily lives.
Besides that, this project also wants to induce the skill to be able to use
technology especially the ICT appropriately and effectively. By using computers to
complete this project, we are introduced to computers and technology which will
enhance our skills of using it. ICT and technologies plays an important role in the
future and to have students who are able to use them wisely and effectively will be an
advantage. Also, this project will also train me to appreciate the intrinsic values of
mathematics and to become more creative and innovative by using different formulas.
This project also makes us realize the importance and beauty of mathematics. When
we are faced with problems, mathematics is able to help us solve it. We also learnt
that there are many ways to solve a problem and this allows us to be more creative.
Mathematics is beautiful because it has many ways to solve a single problem and our
creativity is not limited.
In the process of completing this project, I will learn the importance of
perseverance as time will be invested to the excellence of this project. Similarly, I will
learn the virtue of working together as I will be helping and receiving help from my
fellow peers in the production of this project. Also, I will be thankful and
appreciative. I will be able to apply mathematical principles in daily lives and
appreciate the beauty of mathematics. Besides that, I will also learn to be responsible
because even though we can receive help and work together, the final product still has
to be done by our own.
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5. Question
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6. Section A
Task 1
Materials: 8.5 x 11 inch white paper, 8.5 x 11 inch coloured 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 on the
cylinder. Label it cylinder A.
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7. Take the coloured 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.
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8. Task 2
Dimension Cylinder A Cylinder B
Height 11 8.5
Diameter 2.7 3.5
Radius 1.35 1.75
2. Do you think the two cylinders will hold the same amount?
Do you think one will hold more than the other? Which one? Why?
I don’t think the two cylinders will hold the same amount.
I think Cylinder B will hold more than Cylinder A because it has a larger base
area.
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9. 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 Cylinder A so that
the popcorn falls into Cylinder B. Describe what happened. Is Cylinder B full,
not full, or overflowing?
Cylinder B is not full.
4.
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10. a) Was your prediction correct? How do you know?
b) If your prediction was incorrect, describe what actually happened.
My prediction was correct because Cylinder B can hold more popcorn then
Cylinder A.
5.
a) State the formula for finding the volume of a cylinder.
b) Calculate the volume of Cylinder A?
c) Calculate the volume of Cylinder B?
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.
a) The volume of cylinder is calculated by multiplying the base area and the height of
the cylinder.
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11. b) Volume of Cylinder A =
=
=
c) Volume of Cylinder B =
=
=
d) The volume of the cylinder is different because the volume of a cylinder is affected
more by its radius and not its height.
The radius is squared whereas the height is not, therefore the radius affects the
volume of a cylinder more.
6. Which measurement impacts the volume more: the radius or the height? Work
through the example below to help you answer the questions.
Assume that you have a cylinder with a radius 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 Volume
Original 3 10 282.86
Increased radius 4 10 502.86
Increased height 3 11 311.14
What increased dimension had a larger impact on the volume of the cylinder? Why do
you think this is true?
Volume of original cylinder =
=
=
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12. Volume of cylinder with increased radius =
=
=
Volume of cylinder with increased height =
=
=
The increase in radius of a cylinder has a larger impact on the volume of the cylinder
compared to the increase in the height of a cylinder because the radius is squared.
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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 of thin sheet material.
Explain the details.
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?
1. Cone
The problem is to choose the shape with the largest volume which has a surface area
of .
The strategy is to calculate the maximum volume of a cone with surface area of
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14. 14

15. ( )
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16. 2. Cube
The problem is to choose the shape
with the largest volume which has a
surface area of .
The strategy is to calculate the
maximum volume of a cube with surface
area of
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17. 3. Square-based pyramid
The problem is to choose the shape with the largest volume which has a surface area
of .
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18. The strategy is to calculate the maximum volume of a square-based pyramid with
surface area of
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19. ( )
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20. 20

21. 4. Hemisphere
The problem is to choose the shape with the largest volume which has a surface area
ofategy is to calculate the maximum volume of a hemisphere with surface area of
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22. If I am buying popcorn at the movie theatre and want the most popcorn, I will
look for the hemisphere container because it can hold the most popcorn. I will benefit
from this because I pay less to get more popcorn.
If I am a consumer buying a container, I will but the hemisphere-shaped
container because it had the largest volume and can hold more. Considering that it
uses the same amount of material but can hold the most amounts, it is more worth it
because I am able to store more things with it,
If I am a popcorn seller, I will look for the cube container to sell popcorns.
This is because I will be able to sell more popcorn and earn more profit because I will
need to fill each container with less popcorn compare to the other two containers.
If I am the producer, I will choose to produce the hemisphere container
because I need less material to produce a container with a larger volume. This is
helpful because when I use less material to produce containers with large volume, I
can produce more containers which has the same volume, but still larger than the
other containers made with the same amount of materials. I will be able to earn more
profit.
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23. Conclusion
In Section A, it is proven that a cylinder with a larger radius will have a larger
volume than a cylinder which has a larger height. This means that a cylinder with a
larger diameter will hold more than a cylinder that is taller, given that they have the
same surface area, excluding the two ends.
In Section B, it is proven that a hemisphere shaped container will have the
largest volume compared to a cube shaped and a square-based pyramid shaped
container, given that they have the same surface area, excluding an open end. With a
surface area of ,a hemisphere shaped container has a volume of ,
a cone shaped container has a volume of , a cube shaped container has a
volume of whereas a square-based pyramid shaped container has a
volume of . Therefore for a consumer and the producer, the best container
is the hemisphere shaped container because it has the largest volume compared to
other containers with the same surface area. However, for a popcorn seller the best
container is the cube shaped container because it has the least volume and will be able
to sell more.
From this project, I learned to apply and adapt a variety of problem solving
strategies to solve routine and non-routine problems. I was able to solve the problems
using mathematical strategies and also formulas. This project also increased my
interest and confidence as well as enhances the acquisition of mathematical
knowledge and skills that are useful for career and future undertakings. I also realize
that mathematics is an important and powerful tool in solving real-life problems and
at the same time help me to develop a positive attitude towards mathematics. I was
also able to use ICT and technology appropriately and properly because I was able to
finish this project with the help of computer. I am also trained to appreciate the
intrinsic values of mathematics and to become more creative and innovative by
finding more than one solution towards a problem. Last but not least, this project
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24. helped me to realize the importance and beauty of mathematics because now I have an
insight of how wonderful mathematics can be when used appropriately.
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