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Designing A Water Bottle

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Designing A Water Bottle. W hy do we choose this topic? Students are not willing to bring their own water bottle. T hey always buy the water from the tuck shop . D o not reuse those bottles and just throw them away!  Not environment al friendly!. Objectives.

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Presentation Transcript
slide2

Why do we choose this topic?

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

slide3

Design a water bottle

  • 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.
slide4

h

A

By considering a prism

Fixed

Fixed

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

Fixed

Is the total surface area fixed?

No!

slide5

Perimeter of the base

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

slide6

Our base

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

Parallelogram

Triangle

slide7

Area (A) = bh  h=

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
Right-angled triangle
  • Suppose the area of the triangle is 100 cm 2
right angled triangle1

Conclusion:

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
next consider isosceles triangle

Consider 0o   90o

Area = base (b)  Height (h)

= (2l cos)  (lsin)

1

Length (l)

2

Height (h)

Perimeter (p) = 2l+ b

Base (b)

Next, consider isosceles triangle
result from the graph

Equilateral triangle has the

smallest perimeter

Each angle is 60o ( sum of )

Result from the graph

From the graph, we know that the perimeter is the smallest when

 = 60o

slide15

Parallelogram

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)

slide18

Result from the graph

Rectangle gives the smallest perimeter

  • From the graph,
  • The perimeter is the smallest if  = 90o
slide19

Area (A) = Length (l)  Width(w)

 (where A is fixed)

Perimeter (p) = 2(l + w)

=2 (l + )

Rectangle

Width (w)

Length (l)

slide21

Rectangle

( 10, 40)

slide23

Result from the graph

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

Square gives the smallest perimeter

polygon

Polygon

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

eg.

……

plot p against n if area is fixed
Plot p against n if area is fixed

The perimeter is decreasing as

the number of sides is increasing.

conclusion of the base
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.

slide28

Sphere

Cylinder

Cone

The Bottle

In Form 3, we have learned the solid related circle, they are…

cylinder
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

cylinder1

h

r

Cylinder

Cylinder
  • Suppose the volume of

cylinder is fixed (400cm3), we would like to find the ratio of radius to height so that the surface area is the smallest.

Volume = r2h

Total surface area = 2r2 + 2rh

conclusion of the cylinder

h

r

Cylinder

Conclusion of the cylinder
  • Suppose the volume of the cylinder is fixed, the surface are is the smallest if
slide34

L

h

r

Cone

Cone

Suppose the volume of

cone is fixed (400cm3), we would like to find the ratio of radius to height so that the surface area is the smallest.

Volume = r2h

Total surface area = r2 + rL

conclusion of the cone

h

r

Conclusion of the cone
  • Suppose the volume of the cone is fixed, the surface are is the smallest if
comparison

If r : h = 1: 2

Cylinder

Cone

Volume = r2h

= 2r3

= 6r2

Comparison

r

h = 2r

h = 0.354r

r

r

Sphere

If r : h = 1: 0.353

Volume = r3

Volume = r2h

Surface area = 4r2

= r3

Surface area

= 2rh + 2r2

Surface area

= r

=2.06r2

conclusion
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
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 + 2rh

h

r

cylinder without cover

h

r

Cylinder

Cylinder without cover
  • Suppose the volume of the cylinder is fixed, the surface are is the smallest if
conclusion1
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

slide47

Open-ended Question

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
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

slide49

References :

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/

slide50

END

END

END

slide51

Natural Examples

oranges

watermelons

cherry

calabash

slide52

Chinese Design

Wine container

bowl

slide53

Reflection:

  • 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
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.

slide55

Limitation

  • 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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