Business mathematics

14/05/2015
Business Mathematics
1

Final Presentation
Instructor: Miss Rabia Javed
[ Lecturer Business
Mathematics ]

Group Members
Adeel Iftikhar ID: 20397
Salman Haider ID:20357
Rahul Rai ID: 20400
Jahan Ban Hassan ID:
Abdul Latif ID:
14/05/2015
3

BUSINESS
MATHMETICS
FINAL
PRESENTATION
PARABOLA &
TYPES
& ITS
APPLICATION
IN REAL LIFE

Agenda of today’s
presentation
 Definition & types of parabola.
 History of parabola.
 How to solve parabola.
 Quadratic Equations & function link with
parabola.
 Examples in Engineering.
 Examples in Business.
 Examples regarding Einstein's theory.
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Adeel
Iftikhar
Definition
&
Types

Parabola
A parabola is a two dimensional
curve.
Mirror symmetrical Curve.
Which is approximately U-Shaped.
This parabola represents Quadratic
equation, or Quadratic Functions
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Types of Parabola
Circle
Ellipse
Hyperbola
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Circle
The type of
parabola in :-
• Parabola
facing up
• Parabola
facing down
• Are equal
• & joined with
the same
centre.
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Ellipse
 The case of uneven circle
 It has uneven radius
 Its more of an oval shaped parabola
 In this the parabola facing & down are
joined
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Ellipse
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Hyperbola
like an parabola.
But expanded.
With inverted parabola’s.
Joined parabolas.
With some distance
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12

Hyperbola
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13

All types of Parabola’s
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Edison Works Regarding
Parabola
 Edison first invented the parabola
reflector.
 To focus light on one point.
 It uses used the principle on amplification
as well
 In which parabolic reflector reflected the
light.
 By amplifying light.
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Cont.
 It is the men first step to use light.
 This also gives him a giant leap in the field
of Mirror.
 By knowing their focus property.
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Edison 1st Reflecting Parabola
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17

Parabolic Microphones
 In 21st century.
 Engineers made use of the principle of
Echo sound & parabola.
 To make a device known as Parabolic
Microphones.
 Which helps in collecting the sound.
 From long distances.
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Parabolic Microphone
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19

Construction of Arches
 Construction of arches can be done by
parabolas.
 It construction they balance the forces at
any point.
 So vertex plays a critical role in this.
 Which determines the strength &
 Also the maximum or minimum forces
 The building can handle.
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Arches
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21

Jahan Ban
Hassan
History
Of
Parabola

History
 Parabolas were first made by
Menaechmus in the fourth century BC.
 While solving double cubes.
 The area enclosed by a parabola and a
line segment is called "parabola
segment“.
 Was made by Archimedes.
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History
 The name "parabola" is due to Apollonius.
 Who discovered many properties.
 Conic sections
 Focus property of the parabola and other
conics
 Is due to Pappus.
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History
 Galileo showed that the path of a
projectile
 Follows a parabola.
 & Situation of uniform acceleration
 Due to gravity.
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Parabolas were 1st used in
 Parabolic Reflectors were made due to
parabolas.
 Which then produces Telescopes.
 Isaac Newton built the First Reflecting
Telescope in 1668,
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Equation of Parabola
 The last Equation of parabola is
y = ax2 + bx + c
 Which include:-
 X-intercept
 Y-intercept
 Concavity
 Vertex
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X-Intercept
 The point at which parabola touches X-
axis
 Can be found by putting
y = 0
 But this is optional dependent on
equation.
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Y-Intercept
 The point at which parabola touches Y-
axis
 Can be found by putting
x = 0
 But this is optional dependent on
equation.
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Concavity
Determines the direction of
parabola
May be Concave up or Concave
down.
Which determines either graph is
Rising
or Falling.
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Concavity
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Concave up Concave
Down

Parabola & 21st Century
Parabola have become a
great tool in 21st century.
The domestic devices are
1. Parabolic Solar Panels.
2. Parabolic solar Cookers.
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Parabolic Solar Pressure
Cookers
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33

Parabola & Chemistry
 Now a days, chemist are also using
parabolas.
 Like to compare densities of different
chemicals.
 As density = mass / volume
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Chemistry & Parabola
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Cont.
 The chemical with the higher density
makes parabola.
 The vertex of this chemical is the measure
 Of density
 Which shows how much denser is the
chemical.
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36

Projectile Motion
 A parabolic motion of an object which
works under the constant force of gravity.
 Vertex is measure of minimum
gravitational force at this height.
 This is the result of Newtonian physics.
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Throwing Ball
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Fountain
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Rahul
Rai
14/05/2015
Business Mathematics40
How to
solve
parabola
with
quadratic
equations

How to solve parabola
General equation of parabola
f(x) = y = ax2 + bx + c
Concavity
X- intercept
Y- intercept
Vertex
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1st Concavity
 Of quadratic equation
f(x)= y = ax2 + bx + c
 If a < 0 then parabola is concave down.
 If a > 0 then parabola is concave up.
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2nd X-intercept
f(x)= y = ax2 + bx + c
 Putting y = 0
 We can find value
 By simplifying or if may be
 By using formulas like (a + b)2 ,(a - b)2 ,
(a + b)(a - b)
 We can find the point at which it touches
x-axis
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3rd Y-intercept
f(x)= y = ax2 + bx + c
 By putting x = 0
 Quadratic Equation
 We can find the point at which it touches
y-axis
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4th Vertex
 The maximum point of parabola concave
down (a > 0)
 The minimum point of parabola concave
up (a < 0)
 X = -b/2a
 Y = f(x) = f(-b/2a) or (4ac – b2)/4a
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Beside that
 The general equations of parabola is
Y = a(x - h)2 + k
 Where h & k are vertex points
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Conversion from quadratic
equation to general equation
 y = 2x2 – 4x + 5 into a(x-h)2 + k
 Y = 2(x2 - 2x) + 5 now applying
completing the square method.
 (+)ing & (-)ing by 1
 Y = 2(x2 – 2x + 12 ) – 2 + 5
 Y = 2(x - 1)2 – 2 + 5
 Y = 2(x - 1)2 + 3
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Cont.
 Y = 2(x - 1)2 + 3
 Comparing with Y = a(x - h)2 + k
 x – 1 = x – h
 So h = 1
 K = 3
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Equivalent Equations
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Equivalent Equations
 The equation
Y = 2x2 - 4x + 5
 & also the equation
Y = 2(x - 1)2 + 3
 Are equivalent because they have same
graph.
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Example-1
Y = 4x2
 Comparing with standard quadratic
equation :-
Y = ax2 + bx + c -> (A)
Y = 4x2 -> (B)
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Cont.
By comparing (A) & (B)
a = 4 , b = 0 , c = 0
 Concavity :-
As a > 0, so parabola is concave up-ward
 X-intercept :-
Putting Y = 0 in (B)
Y = 4x2
0 = 4x2
0 = x2
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Cont.
 By taking square root
x = 0 (0,0)
 Y-intercept
Putting x = 0 in (B)
Y = 4(0)2
Y = 4.(0)
Y = 0 (0,0)
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Cont.
 Vertex :-
a = 4 , b = c = 0
As, x = -b/2a
x = -(0)/2.4
x = 0/8
x = 0
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Cont.
 For Y = f(-b/2a) = f(0)
Y = f(0) = 4.(0)2
Y = 4.0
Y = 0 (0,0)
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Graph
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Example-2
Y = -6x2
 Comparing with standard quadratic
equation :-
Y = ax2 + bx + c -> (A)
Y = -6x2 -> (B)
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Cont.
By comparing (A) & (B)
a = 4 , b = 0 , c = 0
 Concavity :-
As a < 0, so parabola is concave down-ward
 X-intercept :-
Putting Y = 0 in (B)
Y = -6x2
0 = -6x2
0 = x2
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Cont.
 By taking square root
x = 0 (0,0)
 Y-intercept
Putting x = 0 in (B)
Y = -6(0)2
Y = -6.(0)
Y = 0 (0,0)
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Cont.
 Vertex :-
a = -6 , b = c = 0
As, x = -b/2a
x = -(0)/2.-6
x = 0/-12
x = 0
14/05/2015
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Cont.
 For Y = f(-b/2a) = f(0)
Y = f(0) = -6.(0)2
Y = -6.0
Y = 0 (0,0)
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Graph
14/05/2015
62

Salman
Haider
14/05/2015
Examples in
Engineering
&
Business

Applications of Parabola:
 The trajectory of a thrown object is a parabolic
curve.
 Compound Interest earned on a savings account is a
parabolic curve.
 Acceleration and deceleration, if graphed, forms a
parabola.
 The surface of water in a rotating container has a
parabolic curve.
 parabolas can be found in lenses, lamps, flashlights,
and lighthouses. when a light source is placed at the
parabola's focus, light emanates in parallel rays,
optimizing its intensity.
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Applications in Business
 Quadratic Equations are used in Business
to find Revenues , Quantity of a product.
 In business it helps in predicting the
maximum or minimum revenue.
 Example :- q = 6p -> (A)
q = Quantity demanded
p = Price of a unit
As, R = p*q
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Cont.
 So (*)ing A by p
p.q = 4.p.p
R = 4p2
 As this is now a quadratic equation.
R = ap2 + bx + c
 By comparing
a = 4 , b = c = 0
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Cont.
 Concavity :- As a > 0
So parabola is concave up-ward.
 R-intercept :- putting p = 0 in Revenue
equation
R = 4.p2
R = 4.(0)2
R = 4.0
R = 0 (0,0)
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Cont.
 P-intercept :- putting q = 0 in revenue
equation
q = 4p2
0 = 4.p2
0 = p2
p = 0 (0,0)
14/05/2015
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Cont.
 Vertex :-
As p = -b/2a
p = -0/2.4
p = 0/8
p = 0
As for R =f(-b/2a) = f(0) = 4.(0)2
R = 4.0
R = 0 (0,0)
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Graph
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70

Solar Energy Farms
 The solar panels which are used in the
generating electricity.
 Have a certain angle.
 Through which they direct there energy to
a central point.
 By which the electricity is generated.
 The parabola has a focus, bouncing all
waves onto a single receiving point
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Solar penal
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Vertex

Cont.
 In construction of solar panels.
 One must remember that the surface
area should be enough.
 To collect maximum amount of solar
energy.
 And one should place its sensor at a
optimum distance
 To collect maximum amount of energy.
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Anatomy of the Solar penal
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Cont.
 In the above figure solar penal are using
focusing property of parabola.
 Under which they get solar energy &
 Then reflect it back towards
evaporator(sensors).
 To get solar energy.
 This energy is then stored into condensors
for future use.
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Parabolas & Lenses
 Parabolas play a vital in
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76

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Abdul
Latif
14/05/2015
Examples in
Engineering
& Einstein's
Theory

Applications in Construction
 Parabola have been used in construction
during 19th century.
 They are used in suspension bridges,
concrete bridges , dams etc.
 The vertex point in construction is the
point.
 At which maximum load can be
supported at a point.
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Examples
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Anatomy of a Suspension
Bridge
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81
Vertex
Axis of Symmetry
(balanced
forces)

Cont.
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Examples Regarding Einstein’s
Theory
Before Einstein’s theory.
It was considered that Earth is
oval.
And sun is circling around the
earth.
So earth is considered the
centre of the universe.
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According to Einstein’s Theory
 That all planets are lying on a same plane.
 This plane is effected
 By the force of Gravity of every planet.
 Planets have their on gravitation to
attract meteor , satellites.
 And also he concluded that sun is centre
of the whole universe.
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Forces Exerted by planet on a
plane
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Force Exerted by Earth
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Vertex of Einstein’s theory
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87
Point of
maximum
force of
earth

Examples in Dish Satellite
 As parabola have a property to direct
force on a single point.
 These principles are used in construction
of Satellite
 So that they can direct their forces to their
central point.
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Parabola’s Focus Property
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89

Satellites using Focus Principle
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90

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91

The End
Business
Mathematics
14/05/2015
Thanks for
your
support
Audience

Business Mathmetics
14/05/2015
93

Business mathematics

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