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
6.7 RationalEqn Apps
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 6.6 ? Rational Equations
• Any QUESTIONS About HomeWork
• 6.6 ? HW-21

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6.7 Rational Equation Applications

• Problems Involving Work
• Problems Involving Motion
• Problems Involving Proportions
• Problems involving Average Cost

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Solve a Formula for a Variable

• Formulas occur frequently as mathematical models.
  Many formulas contain rational expressions, and
  to solve such formulas for a specified letter, we
  proceed as when solving rational equations.

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Solve Rational Eqn for a Variable

1. Determine the DESIRED letter (many times formulas
   contain multiple variables)
2. Multiply on both sides to clear fractions or
   decimals, if that is needed.
3. Multiply if necessary to remove parentheses.
4. Get all terms with the letter to be solved for on
   one side of the equation and all other terms on
   the other side, using the addition principle.
5. Factor out the unknown.
6. Solve for the letter in question, using the
   multiplication principle.

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Example ? Solve for Letter

• Solve this formula for y

• SOLN

Multiplying both sides by the LCD
Simplifying
Multiplying
Subtracting RT
Dividing both sides by Ra
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Example ? Fluid Mechanics

• In a hydraulic system, a fluid is confined to two
  connecting chambers. The pressure in each
  chamber is the same and is given by finding the
  force exerted (F) divided by the surface area
  (A). Therefore, we know

• Solve this Eqn for A2

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Example ? Fluid Mechanics

• SOLUTION

Multiplying both sides by the LCD
Dividing both sides by F1

• This formula can be used to calculate A2 whenever
  A1, F2, and F1 are known

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Problems Involving Work

• Rondae and Marrisa work during the summer
  painting houses.
• Rondae can paint an average size house in 12
  days
• Marrisa requires 8 days to do the same painting
  job.
• How long would it take them, working together,
  to paint an average size house?

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House Painting cont.

• Familiarize. We familiarize ourselves with the
  problem by exploring two common, but incorrect,
  approaches.
• One common, incorrect, approach is to add the two
  times. ? 12 8 20
• Another incorrect approach is to assume that
  Rondae and Marrisa each do half the painting.
• Rondae does ½ in 12 days 6 days
• Marrisa does ½ in 8 days 4 days
• 6 days 4 days 10 days.

?
?
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House Painting cont.

• A correct approach is to consider how much of the
  painting job is finished in ONE day i.e.,
  consider the work RATE
• It takes Rondae 12 days to finish painting a
  house, so his rate is 1/12 of the job per day.
• It takes Marrisa 8 days to do the painting alone,
  so her rate is 1/8 of the job per day.
• Working together, they can complete 1/8 1/12,
  or 5/24 of the job in one day.

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House Painting cont.

• Note That given a TIME-Rate
• Amount RateTimeQuantity

• Form a table to help organize the info

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House Painting cont.

1. Translate. The time that we want is some number t
   for which

Or
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House Painting cont.

1. Carry Out. We can choose any one of the above
   equations to solve

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House Painting cont.

1. Check. Test t 24/5 days

?

1. State. Together, it will take Rondae Marrisa 4
   4/5 days to complete painting a house.

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The WORK Principle

• Suppose that A requires a units of time to
  complete a task and B requires b units of time to
  complete the same task.
• Then A works at a rate of 1/a tasks per unit of
  time.
• B works at a rate of 1/b tasks per unit of
  time,
• Then A and B together work at a rate of 1/a
  1/b per unit of time.

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The WORK Principle

• If A and B, working together, require t units of
  time to complete a task, then their combined rate
  is 1/t and the following equations hold

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Problems Involving Motion

• Because of a tail wind, a jet is able to fly 20
  mph faster than another jet that is flying into
  the wind. In the same time that it takes the
  first jet to travel 90 miles the second jet
  travels 80 miles. How fast is each jet traveling?

r
r20

• HEAD Wind

• TAIL Wind

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HEADwind vs. TAILwind

• Familiarize. We try a guess. If the fast jet is
  traveling 300 mph because of a tail wind the slow
  jet plane would be traveling 300-20 or 280 mph.
• At 300 mph the fast jet would have a 90 mile
  travel-time of 90/300, or 3/10 hr.
• At 280 mph, the other jet would have a
  travel-time of 80/280 2/7 hr.
• Now both planes spend the same amount of time
  traveling, So the guess is INcorrect.

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HEADwind vs. TAILwind

• Translate. Fill in the blanks using
• TimeQuantityDistance/Rate

AirCraft Distance(miles) Speed(miles per hour) Time(hours)
Jet 1 80 r
Jet 2 90 r 20
r
r20
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HEADwind vs. TAILwind

• Set up a RATE Table
• Distance/Rate TimeQuantity

AirCraft Distance(miles) Speed(miles per hour) Time(hours)
Jet 1 80 r 80/r
Jet 2 90 r 20 90/(r 20)
The Times MUST be the SAME
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HEADwind vs. TAILwind

• Since the times must be the same for both planes,
  we have the equation

1. Carry Out. To solve the equation, we first
   Clear-Fractions multiplying both sides by the LCD
   of r(r20)

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HEADwind vs. TAILwind

• Complete the Carry Out

Simplified by Clearing Fractions
Using the distributive law
Subtracting 80r from both sides
Dividing both sides by 10

• Now we have a possible solution. The speed of one
  jet is 160 mph and the speed of the other jet is
  180 mph

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HEADwind vs. TAILwind

1. Check. Reread the problem to confirm that we were
   able to find the speeds. At 160 mph the jet would
   cover 80 miles in ½ hour and at 180 mph the other
   jet would cover 90 miles in ½ hour. Since the
   times are the same, the speeds Chk
2. State. One jet is traveling at 160 mph and the
   second jet is traveling at 180 mph

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Formulas in Economics

• Linear Production Cost Function

• Where
• b is the fixed cost in
• a is the variable cost of producing each unit in
  /unit (also called the marginal cost)
• AverageCost (/unit)

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Formulas in Economics

• Price-Demand Function Suppose x units can be
  sold (demanded) at a price of p dollars per
  units.

• Where
• m n are SLOPE Constants in /unit unit/
• d k are INTERCEPT Constants in units

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Formulas in Economics

• Revenue Function
• Revenue (Price per unit)(No. units sold)

• Profit Function
• Profit (Total Revenue) (Total Cost)

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Example ? Average Cost

• Metro Entertainment Co. spent 100,000 in
  production costs for its off-Broadway play Pride
  Prejudice. Once running, each performance costs
  1000
• Write the Cost Function for conducting z
  performances
• Write the Average Cost Function for the z
  performances
• How many performances, n, result in an average
  cost of 1400 per show

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Example ? Average Cost

• SOLUTION a) Total Cost is the sum of the Fixed
  Cost and the Variable Cost

• SOLUTION b) The Average Cost Fcn

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Example ? Average Cost

• SOLUTION c) In this case

• Thus 250 shows are needed to realize a per-show
  cost of 1400

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Problems Involving Proportions

• Recall that a RATIO of two quantities is their
  QUOTIENT.
• For example, 45 is the ratio of 45 to 100, or
  45/100.
• A proportion is an equation stating that two
  ratios are EQUAL
• An equality of ratios, A/B C/D, is called a
  proportion. The numbers within a proportion are
  said to be proportionAL to each other

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Example ? Triangle Proportions

• Triangles ABC and XYZ are similar

• Note that Similar Triangles are In
  Proportion to Each other

• Now Solve for b if
• x 8, y 12 and a 7

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Example ? Similar Triangles

• SOLUTION - Examine the drawing, write a
  proportion, and then solve.

• Note that side a is always opposite angle A,
  side x is always opposite angle X, and so on.

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Example ? Similar Triangles

• Set Up TheProportions

B
a 7
C
A
b
Y
x 8
Z
X
y 12
b is to 12 as7 is to 8
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Example ? Similar Triangles

• AlternativeProportions

B
a 7
C
A
b
Y
x 8
Z
X
y 12
b is to 7 as12 is to 8
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Example ? Quantity Proportions

• A sample of 186 hard drives contained 4 defective
  drives. How many defective drives would be
  expected in a group of 1302 HDDs?
• Form a proportion in which the ratio of defective
  hard drives is expressed in 2 ways.

• Expect to find 28 defective HDDs

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Whale Proportionality

• To determine the number of humpback whales in a
  pod, a marine biologist, using tail markings,
  identifies 35 members of the pod.
• Several weeks later, 50 whales from the pod are
  randomly sighted. Of the 50 sighted, 18 are from
  the 35 originally identified. Estimate the number
  of whales in the pod.

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Tagged Whale Proportions

• Familarize. We need to reread the problem to
  look for numbers that could be used to
  approximate a percentage of the of the pod
  sighted.
• Since 18 of the 35 whales that were later sighted
  were among those originally identified, the ratio
  18/50 estimates the percentage of the pod
  originally identified.

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HumpBack Whales

1. Translate Stating the Proportion

Original whales sighted later
Whales originally identified
Total Whales sighted later
Entire pod

1. CarryOut

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More On Whales

1. Check. The check is left to the student.
2. State. There are about 97 whales in the Pod

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Example ? Vespa Scooters

• Juans new scooter goes 4 mph faster than Josh
  does on his scooter. In the time it takes Juan
  to travel 54 miles, Josh travels 48 miles.
• Find the speed of each scooter.

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Example ? Vespa Scooters

• Familiarize. Lets guess that Juan is going 20
  mph. Josh would then be traveling 20 4, or 16
  mph.
• At 16 mph, he would travel 48 miles in 3 hr.
  Going 20 mph, Juan would cover 54 mi in 54/20
  2.7 hr. Since 3 ? 2.7, our guess was wrong, but
  we can see that if r the rate, in miles per
  hour, of Juans scooter, then the rate of Joshs
  scooter r 4.

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Example ? Vespa Scooters

• LET
• r Speed of Juans Scooter
• t The Travel Time for Both Scooters
• Tabulate the data for clarity

Distance Speed Time
Juans Scooter
Joshs Scooter
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Example ? Vespa Scooters

• Translate. By looking at how we checked our
  guess, we see that in the Time column of the
  table, the ts can be replaced, using the formula
• Time Distance/Speed

Distance Speed Time
Juans Scooter
Joshs Scooter
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Example ? Vespa Scooters

• Since the Times are the SAME, then equate the
  two Time entries in the table as

• CarryOut

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Example ? Vespa Scooters

• Check If our answer checks, Juans scooter is
  going 36 mph and Joshs scooter is going 36 - 4
  32 mph. Traveling 54 miles at 36 mph, Juan is
  riding for 54/36 or 1.5 hours. Traveling 48
  miles at 32 mph, Josh is riding for 48/32 or 1.5
  hours. The answer checks since the two times are
  the same.
• State Juans speed is 36 mph, and Joshs speed
  is 32 mph

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WhiteBoard Work

• Problems From 6.7 Exercise Set
• 16 (ppt), 34, 44

• Mass Flow Rate for aDivergingNozzle

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P6.7-16

• Given Avg CostFunction Graph

• Find ProductionQuatity for Avg Cost of
  425/Chair
• SOLUTION CastRight Down

20k

• ANS ? 20k Chairs/mon

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All Done for Today
HumanProportions HeadLengthBaseLine
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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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