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## Designing A Water Bottle

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### Polygon

Students are not willing to bring their own water bottle.

They always buy the water from the tuck shop.

Do not reuse those bottles and just throw them away!

Not environmental friendly!

Objectives

- If the volume (V) of the bottle is fixed, we would like to design a water bottle so that its material used (total surface area) is the smallest.

A

By considering a prism

Fixed

Fixed

Volume of the prism = Base area (A) Height (h)

Fixed

Is the total surface area fixed?

No!

h

2

Base area

h

Base area

By considering a prism

- Total surface area = 2 Base area + total areas of lateral faces

= 2 Base area + perimeter of the base height

Conclusion:

The smaller the perimeter of the base is ,

the smaller is the total surface area

First, we need to choose the base for our bottle. We start from the basic figures.

Parallelogram

Triangle

Perimeter (P) = b + h + c

= b + h +

= b + +

c

h

b

Part I: Triangle

First, we begin with a right-angled triangle and assumethe area is fixed.

Right-angled triangle

- Suppose the area of the triangle is 100 cm 2

The perimeter is the smallest if b h

i.e. the right-angled isosceles triangle.

Right-angled triangle- Suppose the area of the triangle is 100 cm 2

Area = base (b) Height (h)

= (2l cos) (lsin)

1

Length (l)

2

Height (h)

Perimeter (p) = 2l+ b

Base (b)

Next, consider isosceles trianglesmallest perimeter

Each angle is 60o ( sum of )

Result from the graphFrom the graph, we know that the perimeter is the smallest when

= 60o

Side (l)

Height (h)

Base (b)

(where 0o 90o)

Area (A) = b h (where A is fixed)

and h= l sin

b =

Perimeter (p) = 2(b + l)

= 2( + l)

(90 , 40)

Rectangle gives the smallest perimeter

- From the graph,
- The perimeter is the smallest if = 90o

Area (A) = Length (l) Width(w)

(where A is fixed)

Perimeter (p) = 2(l + w)

=2 (l + )

Rectangle

Width (w)

Length (l)

( 10, 40)

From the graph, if length = width, the rectangle has the smallest perimeter.

Square gives the smallest perimeter

From the above, we find out that regular figures have the smallest perimeter. So we tried out more regular polygons,

eg.

……

Consider the area of each n-sided polygon is fixed, for example,100cm2.

Conclusion of the base

We know that when the number of sides

Its perimeter

We decided to choose CIRCLEas the base of our water bottle.

Cylinder

Although the volume of the cylinder is fixed, their total surface area are different.

1.5 cm

2.5 cm

3.5 cm

56.6 cm

10.4 cm

20.4 cm

Volume = 400 cm3

Total surface area

= 359.3 cm2

Volume = 400 cm3

Total surface area

= 305.5 cm2

Volume = 400 cm3

Total surface area

= 547.5 cm2

r

Cylinder

Conclusion of the cylinder- Suppose the volume of the cylinder is fixed, the surface are is the smallest if

Cylinder

Cone

Volume = r2h

= 2r3

= 6r2

Comparisonr

h = 2r

h = 0.354r

r

r

Sphere

If r : h = 1: 0.353

Volume = r3

Volume = r2h

Surface area = 4r2

= r3

Surface area

= 2rh + 2r2

Surface area

= r

=2.06r2

If the volume of the 3 solids are fixed, we would like to compare their total surface areas

Conclusion

- From the graph, if the volume is fixed

Surface area of sphere < cylinder < cone

- We know that sphere gives the smallest total surface area.
- However…….

Our choice

The designed bottle is

Cylinder + Hemisphere

In the case the cylinder does

not have a cover. Therefore,

we need to find the ratio of

radius to height of an open

cylinder such that its

surface area is the smallest.

i.e. Total surface area = r2 + 2rh

h

r

Find r : h of cylinder without cover

(1.005 ,238.529)

r

Cylinder

Cylinder without cover- Suppose the volume of the cylinder is fixed, the surface are is the smallest if

Conclusion

From the graph, if r : h = 1: 1, the smallest

surface area of cylinder will be attained.

Volume of bottle = r2(r)+ r3

= r3

E.g. If the volume of water is 500 cm3,

then the radius of the bottle should be 4.57 cm

Can you think of other solids in our daily life / natural environment that have the largest volume but the smallest total surface area?

Member list

School : Hong Kong Chinese Women’s Club College

Supervisor : Miss Lee Wing Har

Group leader: Lo Tin Yau, Geoffrey 3B37

Members: Kwong Ka Man, Mandy 3B09

Lee Tin Wai, Sophia 3B13

Tam Ying Ying, Vivian 3B21

Cheung Ching Yin, Mark 3B29

Lai Cheuk Hay, Hayward 3B36

Book:

Chan ,Leung, Kwok (2001), New Trend Mathematics S3B, Chung Tai Education Press

Website:

http://mathworld.wolfram.com/topics/Geometry.html

http://en.wikipedia.org/wiki/Cone

http://en.wikipedia.org/wiki/Sphere

http://www.geom.uiuc.edu/

- After doing the project, we have learnt :
- more about geometric skills, calculating skills of different prisms, such as cylinders, cones and spheres
- information research and presentation skills
- plotting graphs by using Microsoft Excel
- The most important thing: we learnt that we can use mathematics to explain a lot of things in our daily lives.

Reflection:

Although we faced a lot of difficulties during our project, we never gave up and finally overcame all of them. We widened our horizons and explored mathematics in different aspects in an interesting way.

Also , Miss Lee helped us a lot to solve the difficulties. We would like to express our gratitude and sincere thanks to her.

- Our Maths knowledge is very limited, we wanted to calculate other designs like the calabash or a sphere with a flattened base, but it was to difficult for our level.
- Our knowledge in using Microsoft Excel has caused us a lot of technical problems and difficulties.

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