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8/14/2019 Peka Addmath 09
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-Content-
No. Contents Page
1 Introduction 3 - 42 Objective 5
3 Acknowledge 6
4 Part 1 7 - 12
5 Part 2 13 - 18
7 Part 3 19- 23
8 References 24
-Introduction-
A circle is a simple shape ofEuclidean geometry consisting of those points in a plane which
are the same distance from a given point called thecentre. The common distance of the points of
a circle from its center is called its radius. A diameteris a line segment whose endpoints lie on
http://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpointhttp://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpoint8/14/2019 Peka Addmath 09
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the circle and which passes through the centre of the circle. The length of a diameter is twice the
length of the radius. A circle is never apolygon because it has no sides orvertices.
Circles are simple closed curves which divide the plane into two regions, an interiorand an
exterior. In everyday use the term "circle" may be used interchangeably to refer to either the
boundary of the figure (known as theperimeter) or to the whole figure including its interior, but
in strict technical usage "circle" refers to the perimeter while the interior of the circle is called a
disk. The circumference of a circle is the perimeter of the circle (especially when referring to its
length).
A circle is a special ellipse in which the two foci are coincident. Circles are conic sections
attained when a right circular cone is intersected with a plane perpendicular to the axis of the
cone.
The circle has been known since before the beginning of recorded history. It is the basis for
the wheel, which, with related inventions such asgears, makes much of modern civilization
possible. In mathematics, the study of the circle has helped inspire the development of geometry
and calculus.
Early science, particularly geometry and Astrology and astronomy, was connected to the divine
for most medieval scholars, and many believed that there was something intrinsically "divine" or
"perfect" that could be found in circles.
Some highlights in the history of the circle are:
1700 BC The Rhind papyrus gives a method to find the area of a circular field. The
result corresponds to 256/81 as an approximate value of .[1]
300 BC Book 3 ofEuclid's Elements deals with the properties of circles.
1880 Lindemann proves that is transcendental, effectively settling the millennia-old
problem ofsquaring the circle.[2]
http://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Circle#cite_note-0http://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Circle#cite_note-0http://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-18/14/2019 Peka Addmath 09
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Objectives
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The aims of carrying out this project work are:
1. To apply and adapt a variety of problem-solving strategies to solve
problems;
2. To improve thinking skills;
3. To promote effective mathematical communication;
4. To develop mathematical knowledge through problem solving in a way
that increase students interest and confidence;
5. To use the language of mathematics to express mathematical ideas
precisely;
6. To provide learning environment that stimulates and enhances effective
learning;
7. To develop positive attitude toward mathematics.
Acknowledgement
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First of all I would like to say Alhamdulillah for giving the strength and
health to do this project work.
Not forgotten my parents for providing me everything such as money to
buy anything that are related to this project work and their advice which is the
most needed for this project. Internet , books , computers and all that . They also
supported me and encouraged me to complete this task so that I will not
procrastinate in doing it
Then I would like to thank my teacher Mdm Asmalia Jaafar for guiding
me and my friends throughout this project. We had some difficulties in doing this
task , but she taught us patiently until we know what to do. She tried and tried
to teach us until we understand what we supposed to do with the project work.
Last but not least, my friends who were doing this project with me and
sharing our ideas. They were helpful that when we combined and discussed
together , we had this task done.
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PART 1
SYMBOLS
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Wheel of a bicycle Circles on water surface School park
Fish pond Round table at school compound
Before I continue the task, first, we do have to know what dopi() related to a circle.
Definition
In Euclidean plane geometry, is defined as the ratio of a circle's circumference to its
http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumference8/14/2019 Peka Addmath 09
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diameter:
The ratioC
/d is constant, regardless of a circle's size. For example, if a circle has twice the
diameterdof another circle it will also have twice the circumference C, preserving the ratio C/d.
Area of the circle = area of the shaded square
Alternatively can be also defined as the ratio of a circle's area (A) to the area of a square whose
side is equal to the radius:[3][5]
These definitions depend on results of Euclidean geometry, such
as the fact that all circles aresimilar. This can be considered a
problem when occurs in areas of mathematics that otherwise do
not involve geometry. For this reason, mathematicians often prefer
to define without reference to geometry, instead selecting one of
itsanalytic properties as a definition. A common choice is to
define as twice the smallest positivex for which cos(x) = 0.[6] The formulas below illustrate
other (equivalent) definitions.
History
The ancient Babylonians calculated the area of a circle by taking 3 times the square of its
radius, which gave a value ofpi = 3. One Babylonian tablet (ca. 19001680 BC) indicates a
value of 3.125 forpi, which is a closer approximation.
http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Pi#cite_note-adm-2http://en.wikipedia.org/wiki/Pi#cite_note-4http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-5http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Pi#cite_note-adm-2http://en.wikipedia.org/wiki/Pi#cite_note-4http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-58/14/2019 Peka Addmath 09
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P
B
C
R
Q
A
d1 d2
10 cm
PART 2
Part 2 (a)
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Diagram 1 shows a semicirclePQR of diameter 10cm. SemicirclesPAB andBCR of diameter d1
and d2 respectively are inscribed inPQR such that the sum of d1 and d2 is equal to 10cm. By
using various values of d1 and corresponding values of d2, I determine the relation between
length of arcPQR,PAB, andBCR.
Using formula: Arc of semicircle = d
d1
(cm)
d2
(cm)
Length of arcPQR in
terms of (cm)
Length of arcPAB in
terms of (cm)
Length of arcBCR in
terms of (cm)
1 9 5 9/2
2 8 5 4
3 7 5 3/2 7/2
4 6 5 2 3
5 5 5 5/2 5/2
6 4 5 3 2
7 3 5 7/2 3/2
8 2 5 4
9 1 5 9/2
Table 1
From the Table 1 we know that the length of arcPQR is not affected by the different in d1 and d2
inPAB andBCR respectively. The relation between the length of arcsPQR ,PAB andBCR is
that the length of arcPQR is equal to the sum of the length of arcsPAB andBCR, which is we
can get the equation:
SPQR = SPAB + SBCR
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P R
Q
A
C
d1 d2
10
cm
Bd3
D
E
Let d1= 3, and d2 =7 SPQR = SPAB + SBCR
5 = (3) + (7)
5 = 3/2 + 7/2
5 = 10/2
5 = 5
The Arc length of sector
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d1 d2 d3 SPQR SPAB SBCD SDER1 2 7 5 1/2 7/2
2 2 6 5 3
2 3 5 5 3/2 5/2
2 4 4 5 2 2
2 5 3 5 5/2 3/2
SPQR = SPAB + SBCD + SDER
Let d1 = 2, d2 = 5, d3 = 3 SPQR = SPAB + SBCD + SDER
5 = + 5/2 + 3/2
5 = 5
The length of arc of sector
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15 = 15
The Arc length of sector
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Part 3
a. Area of flower plot = y m2
y = (25/2) - (1/2(x/2)2 + 1/2((10-x )/2)2 )
= (25/2) - (1/2(x/2)2 + 1/2((100-20x+x2)/4) )
= (25/2) - (x2
/8 + ((100 - 20x + x2
)/8) )
= (25/2) - (x2 + 100 20x + x2 )/8
= (25/2) - ( 2x2 20x + 100)/8)
= (25/2) - (( x2 10x + 50)/4)
= (25/2 - (x2 - 10x + 50)/4)
y = ((10x x2)/4)
b. y = 16.5 m2
16.5 = ((10x x2)/4)
66 = (10x - x2) 22/7
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2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 1 2 3 4 5 6 7
X
Y/x
66(7/22) = 10x x2
0 = x2 - 10x + 21
0 = (x-7)(x 3)
x = 7 , x = 3
c. y = ((10x x2)/4)
y/x = (10/4 - x/4)
X 1 2 3 4 5 6 7
y/x 7.1 6.3 5.5 4.7 3.9 3.1 2.4
By using linear law method.
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When x = 4.5 , y/x = 4.3
Area of flower plot = y/x * x
= 4.3 * 4.5
= 19.35m2
Thus, the area of flower plot is equal to 19.35m2
d. Differentiation method
dy/dx = ((10x-x2)/4)
= ( 10/4 2x/4)
0 = 5/2 x/2
5/2 = x/2
x = 5
Completing square method
y = ((10x x2)/4)
= 5/2 - x2/4
= -1/4 (x2 10x)
y+ 52 = -1/4 (x 5)2
y = -1/4 (x - 5)2 - 25
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x 5 = 0
x = 5
e. n = 12, a = 30cm, S12 = 1000cm
S12 = n/2 (2a + (n 1)d
1000 = 12/2 ( 2(30) + (12 1)d)
1000 = 6 ( 60 + 11d)
1000 = 360 + 66d
1000 360 = 66d
640 = 66d
d = 9.697
Thus the diameter of the flower plot is 9.697cm
Tn (flower bed) Diameter
(cm)T1 30
T2 39.697
T3 49.394
T4 59.091
T5 68.788
T6 78.485
T7 88.182
T8 97.879
T9 107.576
T10 117.273T11 126.97
T12 136.667
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REFERENCES
Source via internet :
http://en.wikipedia.org/wiki/Pi
http://ualr.edu/lasmoller/pi.html
http://egyptonline.tripod.com/history.htm
http://www.gap-system.org/~history/HistTopics/Pi_through_the_ages.html
News papers :
News Straits Times
Metro
Berita Harian
Magazines :
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Siswa.
Esti.my