Folio Additional Mathematics

8/6/2019 Folio Additional Mathematics

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Folio ADDiTIONAL

MATHEMATICS

NAME:FAIZAHTUN HUSNA ADNANCLASS:5 IBNU SINA

TEACHER:CIK NOOR AISHAH HAMDAN

NO I/C:940108-08-6150


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Apply mathematics to everyday situation and appreciate the importanceand the beauty of mathematics in everyday lives

To improve problem-solving skills, thinking skills, reasoning andmathematical communication.

To develop positive attitude and personalities and intrinsicmathematical values such as accuracy, confidence and systematicreasoning.

To stimulate learning environment that enhances effective learning,inquiry-based andour team work.

To develop mathematical knowledge in a way which increase ourinterest and confidence.

OBJECTIVE


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FOREWARD


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Part 1


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Every time you graph an equation on a Cartesian coordinate system, you are using the work of

Ren Descartes. Descartes, a French mathematician and philosopher, was born in La Haye,France (now named in his honor) on March 31, 1596. His parents taught him at home until he

was 8 years old, when he entered the Jesuit college of La Flche. There he continued hisstudies until he graduated at age 18.

Descartes was an outstanding student at La Flche, especially in mathematics. Because of hisdelicate health, his teachers allowed him to stay in bed until late morning. Despite missingmost of his morning classes, Descartes was able to keep up with his studies. He would

continue the habit of staying late in bed for his entire adult life.

After graduating from La Flche, Descartes traveled to Paris and eventually enrolled at the

University of Poitiers. He graduated with a law degree in 1616 and then enlisted in a militaryschool. In 1619, he joined the Bavarian army and spent the next nine years as a soldier,

touring throughout much of Europe in between military campaigns. Descartes eventuallysettled in Holland, where he spent most of the rest of his life. There Descartes gave up a

military career and decided on a life of mathematics and philosophy.

Descartes attempted to provide a philosophical foundation for the new mechanistic physicsthat was developing from the work of Copernicus and Galileo. He divided all things into two

categoriesmind and matterand developed a dualistic philosophical system in which,although mind is subject to the will and does not follow physical laws, all matter must obey

the same mechanistic laws.

The philosophical system that Descartes developed, known as Cartesian philosophy, was

based on skepticism and asserted that all reliable knowledge must be built up by the use ofreason through logical analysis. Cartesian philosophy was influential in the ultimate success

of the Scientific Revolution and provides the foundation upon which most subsequentphilosophical thought is grounded.

Descartes published various treatises about philosophy and mathematics. In 1637 Descartes

published his masterwork,Discourse on the Method of Reasoning Well and Seeking Truth inthe Sciences. InDiscourse, Descartes sought to explain everything in terms of matter and

motion.Discourse contained three appendices, one on optics, one on meteorology, and one

titledLa Gometrie (The Geometry). InLa Gometrie, Descartes described what is now

known as the system of Cartesian Coordinates, or coordinate geometry. In Descartes's system

of coordinates, geometry and algebra were united for the first time to create what is known

as analytic geometry.

The Cartesian Coordinate System

Cartesian coordinates are used to locate a point in space by giving its relative distance fromperpendicular intersecting lines. In coordinate geometry, all points, lines, and figures aredrawn in a coordinate plane. By reference to the two coordinate axes, any point, line, or

figure may be precisely located.

In Descartes's system, the first coordinate value (x-coordinate) describes where along the

horizontal axis (the x-axis) the point is located. The second coordinate value (y-coordinate)

locates the point in terms of the vertical axis (the y-axis). A point with coordinates (4, -2) is

located four units to the right of the intersection point of the two axes (point O, or the origin)


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and then two units below the vertical position of the origin. In example (a) of the figure, point

D is at the coordinate location (4, -2). The coordinates for point A are (3, 2); for point B, (2, -4); and for point C, (-2, -5).

The coordinate system also makes it possible to exactly duplicate geometric figures. For

example, the triangle shown in (b) has coordinates A (3,2), B (4, 5), and C (-2, 4) that make it

possible to duplicate the triangle without reference to any drawing.

The triangle may be reproduced by using the coordinates to locate the position of the three

vertex points. The vertex points may then be connected with segments to replicate triangle

ABC. More complex figures may likewise be described and duplicated with coordinates.

A straight line may also be represented on a coordinate grid. In the case of a straight line,every point on the line has coordinate values that must

The

Cartesian coordinate system unites geometry and algebra, and is a universal system for

unambiguous location of points. Applications range from computer animation to global

positioning systems.

satisfy a specific equation. The line in (c) may be expressed as y = 2x. The coordinates of

every point on the line will satisfy the equation y = 2x, as for example, point A (1, 2) andpoint B (2, 4). More complex equations are used to represent circles, ellipses, and curvedlines.

Other Contributions

La Gometrie made Descartes famous throughout Europe. He continued to publish his

philosophy, detailing how to acquire accurate knowledge. His philosophy is sometimes

summed up in his statement, "I think, therefore I am."

Descartes also made a number of other contributions to mathematics. He discovered the Lawof Angular Deficiency for all polyhedrons and was the first to offer a quantifiable

explanation of rainbows. InLa Gometrie, Descartes introduced a familiar mathematicssymbol, a raised number to indicate an exponent. The expression 4 4 4 4 4 may be

written as 45

using Descartes's notation. He also instituted using x, y, and z for unknowns in anequation.

In 1649, Descartes accepted an invitation from Queen Christina to travel to Sweden to be the

royal tutor. Unfortunately for Descartes, the queen expected to be tutored while she did her

exercises at 5:00 A.M. in an unheated library. Descartes had been used to a lifetime of


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sleeping late, and the new routine was much too rigorous for him. After only a few weeks of

this regimen, Descartes contracted pneumonia and died on February 11, 1650.

Geometry and the Fly

Some mathematics historians claim it may be that Descartes's inspiration for the coordinate

system was due to his lifelong habit of staying late in bed. According to some accounts, onemorning Descartes noticed a fly walking across the ceiling of his bedroom. As he watched the

fly, Descartes began to think of how the fly's path could be described without actually tracing

its path. His further reflections about describing a path by means of mathematics led toLa

Gometrie and Descartes's invention of coordinate geometry.

Who Uses Coordinates?

The system of coordinates that Descartes invented is used in many modern applications. For

example, on any map the location of a country or a city is usually given as a set of

coordinates. The location of a ship at sea is determined by longitude and latitude, which is an

application of the coordinate system to the curved surface of Earth. Computer graphic artists

create figures and computer animation by referencing coordinates on the screen.


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Part 2


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Method 1

a)1.Divide area p into 3 segments.2 triangles and 1rectangle.

Triangle 1:=1/2 x 2 x1=1 m2

Triangle 2:=1/2 x 4 x3=6 m2

Rectangle 1:=2 x 4=8 m2

P=(1+6+8)m2P=15 m2

2.Divide area Q into 2 segments.1 triangle and 1rectangle.

Triangle 1:=1/2 x 4 x 3=6 m2


Q=(6 + 6)m2

Q=12 m2

3.Divide area R into 3 segments.1 triangles and 2rectangles.

Triangle 1:=1/2 x 1 x 2=1 m2



R=(1+6+8)m2R=15 m2


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Me ho 2

C i t t t :

077 000 0

(0 6 0 (0 0 8 6 0

= / ( 0

=


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A

0 000 660

= (0 8 8 0 (0 0 0

= (

=


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Me ho 3

Int ti n tA P:

AE = 0

0

AE=

Equation AE

Y=


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mED =

mED=-2

E

ti

ED

Y=-2x + 1

=

=

= =


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Area Q

mAE =

Equation AE

=

= =

=


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=Area R


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Verification answer by GeoGebra:

Area P


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Area Q


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Area R


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b)The mathematics society wishes to fence up the remaining sides of the

region P. Determine the length of the fence required:

from the diagram:


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c) It is impossible for the society to carry out the fencing with an allocation of

RM25 . .

This is because:

RM25 . + RM25/m

=1 m

Therefore,RM25 . can only cover up to 1 m of length

Length of AED B is 12.236m

=12.236 x RM25.

=RM3 5.9 is needed to carry out the fencing plan.

d) i) 1 point for the flag chain to be tied at E

1 point for the flag chain to be tied at a point along the hedge AB

Therefore: 2 points

ii) maximum area of the triangle obtained:

= 9.2 m AE

= 9.2 m 5m

= 4.2 m

Then divide the triangle obtain into 2 right triangle:

Triangle 1

alculate the length.

Using Pythagoras theorem.


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Triangle 2

alculate the length.

Using Pythagoras theorem.

Maximum area of triangle:

=(6 + 2.562 )=8.562

Solution of triangle method:

Angle


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Part 3


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Calculating angle AED by using 2 methods:

Method 1:

Draw a horizontal and vertical lines on Geogebra


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The angle on the left and rightside of AED is calculated:


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l l ti ns:

S ,


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Method 2

Using the formula from the solution of triangle.


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a) Volume of water that has to be pumped in to fill up 80% of the pond.

So,the depth of the pond is 1m.

b) i) The rate of change of depth of the water

ii) The depth of water after 10 minutes


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iii) minimum time taken,inminutes,before the water overflows.

So the water maximum depth of the pond is 1m.Ratio:

iv) the minimum time taken,in minutes,before the water overflows,if the

pond is triangular shaped AED and has a depth of 2 metres.

Now the pond is triangular shaped.Solution of triangle formula can be applied.


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So ratio might work.

Height of water after 10 mins:

Then using ratio,


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Further exploration

a) i)the map in diagram 3

since the scale is 60km per square,we may find the distancebetween x

and the city of Malacca by:

ii)the formula given:

1 nautical mile=1.852 kilometres

difference in latitudes in degreesMalaccas latitude =2Hence,

Differences between the answers.

Yes, there is a different between the answers obtained. This is because the

calculation by using the scale given by a map is only an approximation

method. The answer is correct but less accurate compared to the answer from

the calculation based on the formula given. By using the latitudes, the answer

is very accurate and significant.


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Folio Additional Mathematics