Business mathematics

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Business Mathematics

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Final Presentation

Instructor: Miss Rabia Javed [ Lecturer Business Mathematics ]


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Group MembersAdeel Iftikhar ID: 20397Salman Haider ID:20357Rahul Rai ID: 20400Jahan Ban Hassan ID: Abdul Latif ID:

BUSINESS MATHMETICSFINAL PRESENTATION

PARABOLA & TYPES

& ITS APPLICATIO

NIN REAL

LIFE

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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.

Adeel Iftikhar

Definition&

Types

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ParabolaA 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 ParabolaCircleEllipseHyperbola

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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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Hyperbolalike an parabola.But expanded.With inverted parabola’s. Joined parabolas.With some distance

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Hyperbola

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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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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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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

Jahan Ban Hassan

HistoryOf

Parabola

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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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ConcavityDetermines the direction of

parabolaMay be Concave up or Concave

down.Which determines either graph is

Rising or Falling.

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Concavity

Concave up Concave Down

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Parabola & 21st CenturyParabola have become a great

tool in 21st century.The domestic devices are1. Parabolic Solar Panels.2. Parabolic solar Cookers.

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Parabolic Solar Pressure Cookers

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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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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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Business Mathematics40

Rahul Rai

How to solve

parabola with

quadratic equations

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How to solve parabolaGeneral equation of parabola

f(x) = y = ax2 + bx + cConcavity X- interceptY- interceptVertex

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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 Of quadratic equation

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 Of quadratic equation

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

Þ Y = 2(x2 - 2x) + 5 now applying completing the square method.

Þ (+)ing & (-)ing by 1Þ Y = 2(x2 – 2x + 12 ) – 2 + 5Þ Y = 2(x - 1)2 – 2 + 5Þ 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-interceptPutting 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/2ax = -(0)/2.4x = 0/8x = 0

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Cont. For Y = f(-b/2a) = f(0)

Y = f(0) = 4.(0)2

Y = 4.0Y = 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-interceptPutting 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/2ax = -(0)/2.-6x = 0/-12x = 0

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Cont. For Y = f(-b/2a) = f(0)

Y = f(0) = -6.(0)2

Y = -6.0Y = 0 (0,0)

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Graph

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SalmanHaider

Examples in Engineering

&Business

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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 demandedp = Price of a unit

As, R = p*q

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Cont. So (*)ing A by p

p.q = 4.p.pR = 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

equationq = 4p2 0 = 4.p2 0 = p2 p = 0 (0,0)

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Cont. Vertex :- As p = -b/2a

p = -0/2.4p = 0/8p = 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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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

Vertex

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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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AbdulLatif

Examples in Engineering & Einstein's

Theory

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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 Vertex

Axis of Symmetry(balanced forces)

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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 theoryPoint of maximum force of earth

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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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Satellites using Focus Principle

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The EndBusiness

Mathematics

Thanks for your support Audience

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Business Mathmetics

Business mathematics

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Transcript of Business mathematics