+math project 2009 (:
TRANSCRIPT
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PROJECT WORK FOR
ADDITIONAL MATHEMATHICS 2009
SMK TTDI JAYA
Circles In Our Daily Life
Name : Siti Nurul Idayu bt Abdullah Sapian
Class : 5 Firasat
Teacher : Madam Lee
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Content
No. Contents Page1 Introduction 3
2 Part 1 4 - 6
3 Part 2a 7 - 8
4 Part 2b 9 - 10
5 Part 3 11 - 14
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Introduction
A circle is a simple shape ofEuclidean geometry consisting of thosepoints in aplane which are the
same distance from a given point called the centre. 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 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.
Part 1
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/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpointhttp://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/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/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/Endpointhttp://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_surface -
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There are a lot of things around us related to circles or parts of a circles. We need to play with circles in
order to complete some of the problems involving circles.
amazing circle chives in a circle circle (people)
circle of love circle purple center
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 theratio of a circle'scircumferenceto its
http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Circumferencehttp://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/Circumference -
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diameter:
The ratio
C
/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'sarea (A) to the area of a square whose side is
equal to the radius:
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 its analytic properties as a definition. A common choice is to
define as twice the smallest positivex for whichcos(x) = 0. The
formulas below illustrate other (equivalent) definitions.
History
http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_function -
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d1 d2
10 cm
The circle has been known since before the beginning of recorded history. It is the basis for the wheel,
which, with related inventions such as gears, makes much of modern civilization possible. In
mathematics, the study of the circle has helped inspire the development of geometry and calculus.
Earlyscience, 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 .
300 BC Book 3 ofEuclid's Elementsdeals with the properties of circles.
1880 Lindemann proves that is transcendental, effectively settling the millennia-old problem
ofsquaring the circle.
http://en.wikipedia.org/wiki/Wheelhttp://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/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://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/Squaring_the_circlehttp://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/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circle -
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Part 2 (a)
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, and
BCR.
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
Let d1= 3, and d2 =7 SPQR = SPAB + SBCR
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d1 d2
10
d3D
E
5 = (3) + (7)
5 = 3/2 + 7/2
5 = 10/2
5 = 5
2 (b) i)
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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
(b) ii The length of arc of outer semicircle is equal to the sum of the length of arc of inner semicircle for
n = 1,2,3,4,.
Souter = S1 + S2 + S3 + S4 + S5
(c) Assume the diameter of outer semicircle is 30cm and 4 semicircles are inscribed in the outer
semicircle such that the sum of d1(APQ), d2(QRS), d3(STU), d4(UVC) is equal to 30cm.
d1 d2 d3 d4 SABC SAPQ SQRS SSTU SUVC10 8 6 6 15 5 4 3 3
12 3 5 10 15 6 3/2 5/2 5
14 8 4 4 15 7 4 2 2
15 5 3 7 15 15/2 5/2 3/2 7/2
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let d1=10, d2=8, d3=6, d4=6, SABC = SAPQ + SQRS + SSTU + SUVC
15 = 5 + 4 + 3 + 3
15 = 15
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) - (x
2
+ 100 20x + x
2
)/8
= (25/2) - ( 2x2 20x + 100)/8)
= (25/2) - (( x2 10x + 50)/4)
= (25/2 - (x2 - 10x + 50)/4)
y = ((10x x2)/4)
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3.0
4.0
5.0
6.0
7.0
8.0
0 1 2 3 4 5 6 7
X
Y/x
(b). y = 16.5 m2
16.5 = ((10x x2)/4)
66 = (10x - x2) 22/7
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
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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
(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
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= -1/4 (x2 10x)
y+ 52 = -1/4 (x 5)2
y = -1/4 (x - 5)2 - 25
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
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 117.273
T11 126.97
T12 136.667