W hy 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!
• 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.
Volume of the prism = Base area (A) Height (h)
Fixed
Fixed
Fixed
A
hIs the total surface area fixed?No!
h
Perimeter of the base
h
Base area
Base area
2
• 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.
ParallelogramTriangle
Part I: TrianglePart I: TriangleFirst, we begin with a right-angled triangle and assume the area is fixed.
b
h c2
1
b
A2Area (A) = bh h=
Perimeter (P) = b + h + c
= b + h +
= b + +
22 bh
b
A2 22
2
4b
b
A
Right-angled Right-angled triangletriangle• Suppose the area of the
triangle is 100 cm 2
Base(b)
Perimeter (p)
1 401.0025
2 202.02
: :
14.1 36.92828
: :
Plot p against b, then we find out
perimeter against base if base area is 100 cm square
0
50
100
150
200
250
300
350
400
450
0 20 40 60 80 100 120
base
Perim
eter
perimeter
(14. 1, 36.92828)
Right-angled Right-angled triangletriangle
• 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
: : : : :
Conclusion:The perimeter is the smallest if b h
i.e. the right-angled isosceles triangle.
Next, consider isosceles Next, consider isosceles triangletriangle
Base (b)
Height (h)
Length (l)
Consider 0o 90o
Area = base (b) Height (h)
= (2l cos) (l sin)
2
1
sincos
Al
Perimeter (p) = 2l + b
)cos22(sincos
)cos2(2
A
llp
2
1
Isosceles triangleIsosceles triangleangle in
degreeperimete
r (p)
1302.788
1
2214.119
8: :
6045.5901
4: :
Plot P against the angle Plot perimetre against the angle
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80 90 100
Angle
Perim
etre
perimeter
(60, 45.59014)
Result from the graphFrom the graph, we know that the
perimeter is the smallest when = 60o
Each angle is 60o ( sum of )
Equilateral triangle has the
smallest perimeter
ParallelogramParallelogram
Height (h)
Base (b)
Side (l)
(where 0o 90o)
Perimeter (p) = 2(b + l)
= 2( + l)
Area (A) = b h (where
A is fixed) and h= l sin
b = sinl
A
h
A
sinl
A
angle perimeter
5 249.4798
10 135.1782
: :
60 51.11511
: :
90 40
Parallelogram
Perimeter
0
50
100
150
200
250
300
0 50 100
Perimeter
Parallelogram
(90 , 40)
From the graph,
• The perimeter is the smallest if = 90o
Rectangle gives
the smallest perimet
er
Result from the graph
Length (l)
Width (w)
Area (A) = Length (l) Width(w)
(where A is fixed)
l
Aw
Perimeter (p) = 2(l + w)
=2 (l + ) l
A
Rectangle
lengthperimete
r0.5 4011 202. .. .10 40. .
99.5 201.01100 202
Rectangle
05101520253035404550556065707580859095
100
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
length
perim
eter
( 10, 40)
Rectangle
Area(A) length widthperimeter
100 0.5 200 401
100 1 100 202
. . .. . .
100 10 10 40
. . .
100 99.51.005
03201.0
1
100 100 1 202
Rectangle
From the graph, if length = width, the rectangle has the smallest
perimeter.
Square gives the smallest perimeter
Result from the graph
PolygonPolygonFrom 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.
number of sides (n)
perimeter (p)
4 40
637.22419436
: :
3035.5140933
: :
Plot p against n if area is fixed
perimeter against number of side
35
3637
38
3940
41
0 20 40 60
number of side
perim
eter
perimeter
The perimeter is decreasing as the number of sides is increasing.
Conclusion of the base
We know that when the number of sides
We decided to choose CIRCLE as the base of our
water bottle.
Its perimeter
In Form 3, we have learned the solid related circle, they are…
Cylinder Cone Sphere
Cylinder
1.5 cm2.5 cm
3.5 cm
56.6 cm20.4 cm 10.4 cm
Volume = 400 cm3
Total surface area
= 547.5 cm2
Volume = 400 cm3
Total surface area
= 359.3 cm2
Volume = 400 cm3
Total surface area
= 305.5 cm2
Although the volume of the cylinder is fixed, their total surface area are different.
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 = 2r2 +
2rhCylinder
r
h
Cylinder with coverArea (A) r/h806.2832 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
Plot A against (r/h)
Plot surface area against (r/h)
0
100
200
300
400
500
600
700
800
900
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
r/h
Surface area r/h
(0.5, 300.531)
Conclusion of the cylinder
• Suppose the volume of the cylinder is fixed, the surface are is the smallest if
Cylinder
r
h
2
1
h
r
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
3
1r
Cone
hL
ConeConesurface area r/h610.990626 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
Area against (r/h)
609
609.5
610
610.5
611
611.5
612
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
r/h
Surf
ace
area
r/h
Plot total surface area against (r/h)
(0.353, 609.29)
Conclusion of the cone
• Suppose the volume of the cone is fixed, the surface are is the smallest if
r
h
353.0h
r
Comparison
Cylinder Cone Sphere
If r : h = 1: 0.353
Volume = r2h
Surface area
= 2rh + 2r2
Volume = r2h
If r : h = 1: 2
= 2r3
= 6r2
3
1
= r3 3
353.0
Surface area
= r 22 hr =2.06r2
Volume = r3 3
4
Surface area = 4r2
r r
h = 2r h = 0.354r r
If the volume of the 3 solids are fixed, we would like to compare their total surface
areasvolume
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
: : : :
Compare their surface areas if their volumes are equal
Compare their total surface areas if their volumes equal
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
Surface area
Vol
ume surface(cylinder)
surface(cone)
surface(sphere)
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 choiceThe designed bottle isCylinder + Hemisphere
In the case the cylinder doesnot have a cover. Therefore,we need to find the ratio ofradius to height of an opencylinder such that itssurface area is the smallest.i.e. Total surface area = r2 +
2rh
r
h
Cylinder without coverarea 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
Plot (A) against (r/h)
0
100
200
300
400
500
600
700
800
900
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r/h
tota
l sur
face
are
a(A
)
r/h
Find r : h of cylinder without cover
(1.005 ,238.529)
Cylinder without cover• Suppose the
volume of the cylinder is fixed, the surface are is the smallest if
Cylinder
r
h
1
1
h
r
ConclusionFrom the graph, if r : h = 1: 1, the smallestsurface area of cylinder will be attained.
Volume of bottle = r2(r) + r3
= r3
3
4
2
1
3
5
E.g. If the volume of water is 500 cm3, then the radius of the bottle should be 4.57 cm
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 listSchool : 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
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/
Natural Examples
watermelons
oranges
cherry
calabash
Chinese Design
bowl Wine container
Reflection:After doing the project, we have
learnt :
1.more about geometric skills, calculating skills of different prisms, such as cylinders, cones and spheres
2.information research and presentation skills
3.plotting graphs by using Microsoft Excel
4.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.
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.