Designing A Water Bottle

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Designing A Water Bottle
2
Objectives

• 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!

3
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.

4
By considering a prism
Fixed
Fixed

• Volume of the prism Base area (A) ? Height
  (h)

Fixed
Is the total surface area fixed?
No!
5
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
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Our base
First, we need to choose the base for our bottle.
We start from the basic figures.
Parallelogram
Triangle
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Part I Triangle
First, we begin with a right-angled triangle and
assume the area is fixed.
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Right-angled triangle

• Suppose the area of the triangle is 100 cm 2

Base(b) Perimeter (p)
1 401.0025
2 202.02

14.1 36.92828

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Plot p against b, then we find out
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Right-angled triangle

• Suppose the area of the triangle is 100 cm 2

Area(A) Base(b) Height(h) Hypotenuse(a) Perimeter (p)
100 1 200 200.0025 401.0025
100 2 100 100.02 202.02

100 14.1 14.1844 20.00204 36.92828

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Next, consider isosceles triangle
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Isosceles triangle
angle ? in degree perimeter (p)
1 302.7881
2 214.1198

60 45.59014

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Plot P against the angle ?
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Result from the graph

• From the graph, we know that the perimeter is the
  smallest when
• ? 60o

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Parallelogram

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

and h l sin? b
Perimeter (p) 2(b l)
2( l)
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Parallelogram
angle ? perimeter
5 249.4798
10 135.1782

60 51.11511

90 40
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Parallelogram
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Result from the graph

• From the graph,
• The perimeter is the smallest if ? 90o

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Rectangle
Width (w)
Length (l)
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Rectangle
length perimeter
0.5 401
1 202
. .
. .
10 40
. .
99.5 201.01
100 202
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Rectangle
( 10, 40)
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Rectangle
Area(A) length width perimeter
100 0.5 200 401
100 1 100 202
. . .
. . .
100 10 10 40
. . .
100 99.5 1.00503 201.01
100 100 1 202
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Result from the graph
From the graph, if length width, the rectangle
has the smallest perimeter.
Square gives the smallest perimeter
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Polygon

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

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Consider the area of each n-sided polygon is
fixed, for example,100cm2.
number of sides (n) perimeter (p)
4 40
6 37.22419436

30 35.5140933

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Plot p against n if area is fixed
The perimeter is decreasing as the number of
sides is increasing.
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Conclusion of the base

• We know that when the number of sides

Its perimeter
We decided to choose CIRCLE as the base of our
water bottle.
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The Bottle

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

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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
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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
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Cylinder with cover
Area (A) r/h
806.2832 0.007854
734.8754 0.010454
675.7145 0.013572
626.0032 0.017255
583.7436 0.021551
547.4705 0.026507
516.085 0.03217
488.7466 0.038587
464.802 0.045804
443.7349 0.05387
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Plot A against (r/h)
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Conclusion of the cylinder

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

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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
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Cone
surface area r/h
610.990626 0.404648
610.785371 0.401299
610.594731 0.397996
610.418393 0.394738
610.256057 0.391525
610.107426 0.388354
609.972212 0.385227
609.850136 0.382141
609.740922 0.379096
609.644304 0.376091
609.560021 0.373126
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Plot total surface area against (r/h)
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Conclusion of the cone

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

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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
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If the volume of the 3 solids are fixed, we would
like to compare their total surface areas
volume Surface (cylinder) surface(cone) surface(sphere)
100 119.2654 270.2723782 104.1879416
200 189.3221 429.0306575 165.3880481
300 248.0821 562.1892018 216.7196518

1000 553.581 1254.493253 483.5975862
1100 589.8972 1336.790816 515.3226696

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Compare their surface areas if their volumes are
equal
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Conclusion

• From the graph, if the volume is fixed
• Surface area of sphere lt cylinder lt cone
• We know that sphere gives the smallest total
  surface area.
• However.

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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
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Cylinder without cover
area r/h
803.1415 0.007854
731.0739 0.010453
671.1904 0.013571
620.6938 0.017255
577.5859 0.021551
540.4017 0.026506
508.0422 0.032169
479.6672 0.038585
454.6229 0.045803
432.3934 0.053869
412.566 0.06283
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Find r h of cylinder without cover
(1.005 ,238.529)
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Cylinder without cover

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

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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
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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?
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Member list
School Hong Kong Chinese Womens 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

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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 h
ttp//en.wikipedia.org/wiki/Sphere http//www.geom
.uiuc.edu/
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END
END
END
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Natural Examples
oranges
watermelons
cherry
calabash
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Chinese Design
Wine container
bowl
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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.

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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.

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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.