Add Math Project 2/2012

Form 4

Name: Kuek Yuen MengClass: 4WMycard number: 961115-14-6409School: Wesley Methodist School Kuala Lumpur

Teachers name: Mr. SivaContentItemPage number

Content2

Questions3 - 6

Introduction7 - 8

Section A9 - 15

Section B16 - 24

Conclusion25

Question

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. 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. 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 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 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 learn how to solve a problem correctly. Many careers are also related heavily to mathematics and if we have the necessary mathematical skills, we will be able to excel in them. 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.

Section ATask 1

Materials: 8.5 x 11 in. white paper, 8.5 x 11 in. 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.

2. 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.

Task 2

DimensionCylinder ACylinder B

Height118.5

Diameter2.73.5

Radius1.351.75

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 dont 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.3. Place Cylinder B on the paper plate with Cylinder A inside it. Use your sup 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.

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.

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.CylinderRadiusHeightVolume

Original310282.86

Increased radius410502.86

Increased height311311.14

What increased dimension had a larger impact on the volume of the cylinder? Why do you think this is true?

Volume of original cylinder =

=

= 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.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. ConeThe 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

()

ii. 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

iii. Square-based pyramid

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 square-based pyramid with surface area of

()

iv. Hemisphere

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 hemisphere with surface area of

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.

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 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.25