W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
6.8 Modelby Variation
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 6.7 ? Formulas and Applications of Rational
  Equations
• Any QUESTIONS About HomeWork
• 6.7 ? HW-22

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6.8 Direct and Inverse Variation

• Equations of Direct Variation
• Problem Solving with Direction Variation
• Equations of Inverse Variation
• Problem Solving with Inverse Variation

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Direct Variation

• Many problems lead to equations of the form y
  kx, where k is a constant. Such eqns are called
  equations of variation
• DIRECT VARIATION ?

When a situation translates to an equation
described by y kx, with k a constant, we say
that y varies directly as x. The equation y kx
is called an equation of direct variation.
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Variation Terminology

• Note that for k gt 0, any equation of the form y
  kx indicates that as x increases, y increases as
  well
• Synonyms
• y varies as x,
• y is directly proportional to x,
• y is proportional to x
• The Synonym Terms also imply direct variation and
  are often used

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The Constant k

• For the Direct Variation Equation

• The constant k is called the constant of
  proportionality or the variation constant.
• k can be found if one pair of values for x and y
  is known.
• Once k is known, other (x,y) pairs can be
  determined

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Example ? Direct Variation

• If y varies directly as x, and y 3 when x
  12, then find the eqn of variation
• SOLUTION The words y varies directly as x
  indicate an equation of the form y kx

• Solving for k

• Thus the Equation of Variation

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Example ? Direct Variation cont.

• Graphing the Equation of Variation

• Direct Variation Always produces a SLANTED LINE
  that Passes Thru the ORIGIN

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Example ? Direct Variation

• Find an equation in which a varies directly as b,
  and a 15 when b 25.
• Find the value of a when b 36
• SOLUTION

• Thus the Variation Eqn

• Sub b 36 into Eqn

• Thus when b 36, then the value of a is 21-3/5

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Example ? Bolt Production

• The number of bolts B that a machine can make
  varies directly as the time T that it operates.
• The machine makes 3288 bolts in 2 hr
• How many bolts can it make in 5 hr
• Familarize and Translate The problem states
  that we have DIRECT VARIATION between B and T.
• Thus an equation B kT applies

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Example ? Bolt Production cont.1

1. Carry Out

• Solve for k

• Thus the Equation of Variation

• If T 5 hrs

• Note that k is a RATE with UNITS

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Example ? Fluid Statics

• The pressure exerted by a liquid at given point
  varies directly as the depth of the point beneath
  the surface of the liquid.
• If a certain liquid exerts a pressure of 50
  pounds per square foot (psf) at a depth of 10
  feet, then find the pressure at a depth of 40
  feet.
• SOLN Another case of Direct Variation

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Example ? Fluid Statics
(units are lb/ft3)
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Example ? Fluid Statics

• Use k 5 lb/ft3 in the Direct Variation Equation
  to find the pressure at a depth of 40ft

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Inverse Variation

• INVERSE VARIATION ?

When a situation translates to an equation
described by y k/x, with k a constant, we Say
that y varies INVERSELY as x. The equation y
k/x is called an equation of inverse variation

• Note that for k gt 0, any equation of the form y
  k/x indicates that as x increases, y decreases

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Example ? Inverse Variation

• If y varies inversely as x, and y 30 when x
  20, find the eqn of variation
• SOLUTION The words y varies inversely as x
  indicate an equation of the form y k/x

• Solving for k

• Thus the Equation of Variation

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Example ? Barn Building

• It takes 56 hours for 25 people to raise a barn.
  
• How long would it take 35 people to complete
  the job?
• Assume that all people are working at the same
  rate.

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Example ? Barn Building cont.1

• Familarize. Think about the situation. What kind
  of variation applies? It seems reasonable that
  the greater number of people working on a job,
  the less time it will take. So LET
• T the time to complete the job, in hours,
• N the number of people working
• Then as N increases, T decreases and inverse
  variation applies

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Example ? Barn Building cont.2

1. Translate Since inverse variation applies use

1. Carry Out Find the Constant of Inverse
   Proportionality

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Example ? Barn Building cont.3

1. Carry Out The Eqn of Variation

• When N 35. Find T

1. Chk A check might be done by repeating the
   computations or by noting that (28.8)(35) and
   (56)(25) are both 1008.

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Example ? Barn Building cont.4

1. STATE if It takes 56 hours for 25 people to
   raise a barn, then it should take 35 people
   about 29 hours to build the same barn

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To Solve Variation Problems

1. Determine from the language of the problem
   whether direct or inverse variation applies.
2. Using an equation of the form y kx for direct
   variation or y k/x for inverse variation,
   substitute known values and solve for k.
3. Write the equation of variation and use it, as
   needed, to find unknown values.

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Applications Tips ReDux

• The Most Important Part of Solving REAL WORLD
  (Applied Math) Problems

Translating

• The Two Keys to the Translation
• Use the LET Statement to ASSIGN VARIABLES
  (Letters) to Unknown Quantities
• Analyze the RELATIONSHIP Among the Variables and
  Constraints (Constants)

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Solving Variation Problems

1. Write the equation with the constant of
   variation, k.
2. Substitute the given values of the variables into
   the equation in Step 1 to find the value of the
   constant k.
3. Rewrite the equation in Step 1 with the value k
   from Step 2
4. Use the equation from Step 3 to answer the
   question posed in the problem.

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Other Variation Relations

• Some Additional Variation Eqns
• y varies directly as the nth power of x if there
  is some positive constant k such that

• y varies inversely as the nth power of x if there
  is some positive constant k such that

• y varies jointly as x and z if there is some
  positive constant k such that.

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Combined Variation

• The Previous Variation Forms can be combined to
  create additional equations
• z varies directly as x and INversely as y if
  there is some positive constant k such that

• w varies jointly as x y and inversely as z to
  the nth power if there is some positive constant
  k such that

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Example ? Luminance

• The Luminance of a light (E) varies directly with
  the intensity (I) of the light source and
  inversely with the square distance (D) from the
  light. At a distance of 10 feet, a light meter
  reads 3 units for a 50-cd lamp. Find the
  Luminance of a 27-cd lamp at a distance of 9
  feet.
• This is a case of COMBINED Variation ?

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Example ? Luminance

• Solve for the Variation Constant, k,Using the
  KNOWN values of I D

• Use the value of k, and D 9ft in the variation
  eqn to find E(9ft)

• State At 9ft the 27cd Lamp produces a Luminance
  of 2 cd/m2

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Example ? Sphere Volume

• Suppose that you had forgotten the formula for
  the volume of a sphere, but were told that the
  volume V of a sphere varies directly as the cube
  of its radius r. In addition, you are given that
  V 972p when r 9in.
• Find V when r 6in
• SOLUTION Recognize as Direct Variation to a
  Power V kr3

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Example ? Sphere Volume

• Now use KNOWN data to solve for k

• Now Substitute k 4p/3 into the Eqn of
  Variation

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Example ? Sphere Volume

• Finally Substitute r 6 and solve for V(6) as
  requested

• Using p 3.14159 find the Volume for a 6 inch
  radius sphere, V(6) 904.78 in3

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Example ? Newtons Law

• Newtons Law of Universal Gravitation says that
  every object in the universe attracts every other
  object with a force acting along the line of the
  centers of the two objects and that this
  attracting force is directly proportional to the
  product of the two masses and inversely
  proportional to the square of the distance
  between the two objects.

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Example ? Newtons Law

• Write the Gravitation Law Symbolically
• SOLUTION Let m1 and m2 be the masses of the two
  objects and r be the distance between them a
  Diagram

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Example ? Newtons Law

• Next LET
• F the gravitational force between the objects
• G the Constant of Variation a.k.a., the
  constant of proportionality
• Thus Newton Gravitation Law in Symbolic form

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Example ? Newtons Law

• The constant of proportionality G is called the
  universal gravitational constant. It is termed a
  universal constant because it is thought to be
  the same at all places and all times and thus it
  universally characterizes the intrinsic strength
  of the gravitational force.
• If the masses m1 and m2 are measured in
  kilograms, r is measured in meters, and the force
  F is measured in newtons, then the value of G

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Example ? Newtons Law

• Next Estimate the value of g the acceleration
  due to gravity near the surface of the Earth.
  Use these estimates
• Radius of Earth RE 6.38 x 106 meters
• Mass of the Earth ME 5.98 x 1034 kg
• SOLUTION By Newtons 1st Law Force
  (mass)(acceleration) ?

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Example ? Newtons Law

• Now the Force of Gravity at the earths surface
  is the result of the Acceleration of Gravity

• Equating the Force of Gravity and the
  Gravitation Force Equations

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Example ? Newtons Law

• CarryOut

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Example ? Kinetic Energy

• The kinetic energy of an object varies directly
  as the square of its velocity.
• If an object with a velocity of 24 meters per
  second has a kinetic energy of 19,200 joules,
  what is the velocity of an object with a kinetic
  energy of 76,800 joules?
• SOLUTION This is case of Direct Variation to the
  Power of 2

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Example ? Kinetic Energy

• Write the Equation of Variation

• Next Solve for the Variation Constant, k, using
  the known data

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Example ? Kinetic Energy

• To find k, use the fact that an object with a
  velocity of 24 m/s has a kinetic energy of 19.2 kJ

• Thus k 33.33 J/m2

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Example ? Kinetic Energy

• Use k 33.33 J/m2 to refine the Variation
  Equation

• Next use the E(v) eqn to find v for E 76.8 kJ2

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Example ? Kinetic Energy

• The v for E 76.8 kJ

• Thus when E 76.8 kJ the velocity is 48 m/s

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WhiteBoard Work

• Problems From 6.8 Exercise Set
• 33, 38

• KINETIC andPOTENTIALEnergyBalance

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All Done for Today
Heat FlowsHot?Cold
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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

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Graph y x

• Make T-table

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