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
3.2a SystemApplications
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• s3.1 ? Systems of Linear Equations
• Any QUESTIONS About HomeWork
• s3.1 ? HW-08

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System Methods Compared

• We now have three distinctly different ways to
  solve a system. Each method has strengths
  weaknesses

Method Strengths Weaknesses
Graphical Solutions are displayed visually. Works with any system that can be graphed. Inexact when solutions involve numbers that are not integers or are very large and off the graph.
Substitution Always yields exact solutions. Easy to use when a variable is alone on one side of an equation. Introduces extensive computations with fractions when solving more complicated systems. Solutions are not graphically displayed.
Elimination Always yields exact solutions. Easy to use when fractions or decimals appear in the system. Solutions are not graphically displayed.
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Solving Application Problems

• Read the problem as many times as needed to
  understand it thoroughly. Pay close attention to
  the questions asked to help identify the quantity
  the variable(s) should represent. In other Words,
  FAMILIARIZE yourself with the intent of the
  problem
• Often times performing a GUESS CHECK operation
  facilitates this Familiarization step

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Solving Application Problems

• Assign a variable or variables to represent the
  quantity you are looking for, and, when
  necessary, express all other unknown quantities
  in terms of this variable. That is, Use at LET
  statement to clearly state the MEANING of all
  variables
• Frequently, it is helpful to draw a diagram to
  illustrate the problem or to set up a table to
  organize the information

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Solving Application Problems

1. Write an equation or equations that describe(s)
   the situation. That is, TRANSLATE the words into
   mathematical Equations
2. Solve the equation(s) i.e., CARRY OUT the
   mathematical operations to solve for the assigned
   Variables
3. CHECK the answer against the description of the
   original problem (not just the equation solved in
   step 4)

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Solving Application Problems

1. Answer the question asked in the problem. That
   is, make at STATEMENT in words that clearly
   addressed the original question posed in the
   problem description

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Example ? Problem Solving

• Two angles are supplementary. One of the angles
  is 20 larger than three times the other. Find
  the two angles
• Familarize
• Recall that two angles are supplementary if the
  sum of their measures is 180. We could try and
  guess, but instead lets make a drawing and
  translate. Let x and y represent the measures
  of the two angles

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Example ? Problem Solving

• Familarize with Diagram

y
x

• Translate
• Since the angles are supplementary, one equation
  is x y 180 (1)
• The second sentence can be rephrased and
  translated as follows

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Example ? Problem Solving

• Translating
• Rewording and Translating

One angle is 20? more than three times the other
y 20 3x
(2)

• We now have a system of two equations and two
  unknowns.

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Example ? Problem Solving

• Carry Out

• Sub x 40 in Eqn-1

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Example ? Problem Solving

• Check
• If one angle is 40 and the other is 140, then
  the sum of the measures is 180. Thus the angles
  are supplementary. If 20 is added to three
  times the smaller angle, we have 3(40) 20
  140, which is the measure of the other angle.
  The numbers check.
• STATE
• One angle measures 40 and the other measures
  140

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Elimination Applications

• Total-Value Problems
• Mixture Problems

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Total Value Problems

• EXAMPLE Lupe sells concessions at a local
  sporting event.
• In one hour, she sells 72 drinks. The drink
  sizes are
• small, which sells for 2 each
• large, which sells for 3 each.
• If her total sales revenue was 190, how many of
  each size did she sell?

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Example ? Drinks Sold

• Familiarize.
• Suppose (i.e., GUESS) that of the 72 drinks, 20
  where small and 52 were large.
• The 72 drinks would then amount to a total of
  20(2) 52(3) 196.
• Although our guess is incorrect (but close),
  checking the guess has familiarized us with the
  problem.

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Example ? Drinks Sold

• Familiarize LET
• s the number of small drinks and
• l the number of large drinks
• Translate.
• Since a total of 72 drinks were sold, we must
  have s l 72.
• To find a second equation, we reword some
  information and focus on the income from the
  drinks sold

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Example ? Drinks Sold

• Translate.
• Translating Rewording

Income fromsmall drinks
Income fromlarge drinks
Totals
190
Plus
2s

3l

190

• Thus we haveConstructedthe System

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Example ? Drinks Sold

1. Carry Out

Solve (1) for l
Use (3) to Sub for l in (2)
Use Distributive Law
Combine Terms
Simplify to find s

• Sub s 26 in (3) to Find l

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Example ? Drinks Sold

• Check If Lupe sold 26 small and 46 large drinks,
  she would have sold 72 drinks, for a total of
• 26(2) 46(3) 52 138 190
• State Lupe sold
• 26 small drinks
• 46 large drinks

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Problem-Solving TIP

• When solving a problem, see if it is patterned or
  modeled after a problem that you have already
  solved.

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Example ? Problem Solving

• A cookware consultant sells two sizes of pizza
  stones. The circular stone sells for 26 and the
  rectangular one sells for 34. In one month she
  sold 37 stones. If she made a total of 1138
  from the sale of the pizza stones, how many of
  each size did she sell?

Pizza Stone
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Example ? Pizza Stones

• Familiarize When faced with a new problem, it
  is often useful to compare it to a similar
  problem that you have already solved. Here
  instead of 2 and 3 drinks, we are counting 26
  34 pizza stones. So LET
• c the no. of circular stones
• r the no. of rectangular stones

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Example ? Pizza Stones cont.2

• Translate Since a total of 37 stones were sold,
  we have c r 37
• Tabulating the Data Can be Useful

Total
Rectangular
Circular
34
26
Cost per pan
37
r
c
Number of pans
1138
34r
26c
Money Paid
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Example ? Pizza Stones

• Translate We have translated to a system of
  equations
• c r 37 (1)
• 26c 34r 1138 (2)
• Carry Out ? Multiply Eqn(1) by -26

• Add Eqns (2)(3)

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Example ? Pizza Stones cont.4

• Carry Out Solve for r

• Find c using Eqn (1)

1. Check If r 22 and c 15, a total of 37
   stones were sold. The amount paid was 22(34)
   15(26) 1138 ?
2. State The consultant sold 15 Circular and 22
   rectangular pizza stones

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Example ? Mixture Problem

• A Chemical Engineer wishes to mix a reagent that
  is 30 acid and another reagent that is 50 acid.
  
• How many liters of each should be mixed to get 20
  L of a solution that is 35 acid?

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Example ? Acid Mixxing

• Familiarize. Make a drawing and then make a
  guess to gain familiarity with the problem
• The Diagram

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Example ? Acid Mixxing

• To familiarize ourselves with this problem, guess
  that 10 liters of each are mixed. The resulting
  mixture will be the right size but we need to
  check the Pure-Acid Content

• Our 10L guess produced 8L of pure-acid in the
  mix, but we need 0.35(20) 7L of pure-acid in
  the mix

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Example ? Acid Mixxing

• Translate
• LET
• t the number of liters of the 30 soln
• f the number of liters of the 50 soln
• Next Tabulate the calculation of the amount of
  pure-acid in each of the mixture components

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Example ? Acid Mixxing

• Pure-Acid Calculation Table

• The Table Reveals a System of Equations

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Example ? Acid Mixxing

1. Carry Out Solve Eqn System

• Eliminate f by multiplying both sides of equation
  (1) by -0.5 and adding them to the corresponding
  sides of equation (2)

0.50t 0.50f 10
0.30t 0.50f 7
0.20t 3
t 15.
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Example ? Acid Mixxing

• To find f, we substitute 15 for t in equation
  (1) and then solve for f

15 f 20
f 5

• Obtain soln (15, 5), or t 15 and f 5
• Check Recall t is the of liters of 30 soln and
  f is the of liters of 50 soln

?
Number of liters t f 15 5 20
Amount of Acid 0.30t 0.50f
0.30(15) 0.50(5) 7
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Example ? Wage Rate

• Ethan and Ian are twins. They have decided to
  save all of the money they earn at their
  part-time jobs to buy a car to share at college.
  One week, Ethan worked 8 hours and Ian worked 14
  hours. Together they saved 256. The next week,
  Ethan worked 12 hours and Ian worked 16 hours and
  they earned 324.
• How much does each twin make per hour?
• i.e. What are the Wage RATES

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Example ? Wage Rate

• In This Case LET
• E Ethans Wage Rate (/hr)
• I Ians Wage Rate (/hr)
• Translate Ethan worked 8 hours and Ian worked 14
  hours. Together they saved 256
• 8Ethans Rt plus 14Ians Rt is 256

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Example ? Wage Rate

• Translate Ethan worked 12 hours and Ian worked
  16 hours and they earned 324.
• 12Ethans Rt plus 16Ians Rt is 324

• Now have 2-Eqn System

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Example ? Wage Rate

• Carry Out

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Example ? Wage Rate

• Carry Out

• Sub I 12 into 1st eqn to Find E

• State Answer
• Ethan Earns 11 per hour
• Ian Earns 12 per hour

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Example ? Geometry

• The perimeter of a fence around the childrens
  section of the community park is 268 feet. The
  length is 34 feet longer than the width. Find the
  dimensions of the park.
• Familiarize Draw a Diagram and LET
• l the length
• w the width

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Example ? Geometry

• Translate.
• The Perimeter is 2l 2w.
• The perimeter is 268 feet.
• 2l 2w 268
• The length is 34 ft more than the width
• l 34 w

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Example ? Geometry

• Now have a system of two equations and two
  unknowns.
• 2l 2w 268 (1)
• l 34 w (2)
• Solve for wusingSubstitutionMethod

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Example ? Geometry

• Sub 50 for w in one of the Orignal Eqns

l 34 w 34 50 84 feet

1. Check If the length is 84 and the width is 50,
   then the length is 34 feet more than the width,
   and the perimeter is 2(84) 2(50), or 268 feet
   
2. State The width is 50 feet and the length is 84
   feet

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Example ? Mixture Problem

• A coffee shop is considering a new mixture of
  coffee beans. It will be created with Italian
  Roast beans costing 9.95 per pound and the
  Venezuelan Blend beans costing 11.25 per pound.
  The types will be mixed to form a 60-lb batch
  that sells for 10.50 per pound.
• How many pounds of each type of bean should go
  into the blend?

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Example ? Coffee Beans

• Familiarize This problem is similar to one of
  the previous examples.
• Instead of pizza stones we have coffee beans
• We have two different prices per pound.
• Instead of knowing the total amount paid, we know
  the weight and price per pound of the new blend
  being made.
• LET
• i no. lbs of Italian roast and
• v no. lbs of Venezuelan blend

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Example ? Coffee Beans

• Translate Since a 60-lb batch is being made, we
  have i v 60.
• Present the information in a table.

Italian Venezuelan New Blend
Number of pounds i v 60
Price per pound 9.95 11.25 10.50
Value of beans 9.95i 11.25v 630
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Example ? Coffee Beans cont.4

1. Translate - We have translated to a system of
   equations

• Carry Out - When equation (1) is solved for v, we
  have v 60 - i.
• We then substitute for v in equation (2).

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Example ? Coffee Beans cont.5

1. Carry Out - Find v using v 60 - i.

• Check - If 34.6 lb of Italian Roast and 25.4 lb
  of Venezuelan Blend are mixed, a 60-lb blend will
  result.
• The value of 34.6 lb of Italian beans is
  34.6(9.95), or 344.27.
• The value of 25.4 lb of Venezuelan Blend is
  25.4(11.25), or 285.75,

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Example ? Coffee Beans cont.6

• Check cont.
• so the value of the blend is 344.27 285.75
  630.02.
• A 60-lb blend priced at 10.50 a pound is also
  worth 630, so our answer checks
• State The blend should be made from
• 34.6 pounds of Italian Roast beans
• 25.4 pounds of Venezuelan Blend beans

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Simple Interest

• If a principal of P dollars is borrowed for a
  period of t years with interest rate r (expressed
  as a decimal) computed yearly, then the total
  interest paid at the end of t years is

• Interest computed with this formula is called
  simple interest. When interest is computed
  yearly, the rate r is called an annual interest
  rate

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Example ? Simple Interest

• Ms. Jeung invests a total of 10,000 in Bonds
  from blue-chip and technology Companies. At the
  end of a year, the blue-chips returned 12 and
  the technology stocks returned 8 on the original
  investments.
• How much was invested in each type of Bond if the
  total interest earned was 1060?

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Example ? Simple Interest

• Familiarize
• We are asked to find two amounts
• that invested in blue-chip Bonds
• that invested in technology Bonds.
• If we know how much was invested in blue-chip
  Bonds, then we know that the rest of the 10,000
  was invested in technology Bonds.

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Example ? Simple Interest

• TRANSLATE LET
• x amount invested in blue-chip bonds.
• y amount invested in tech bonds
• Tabulate Interest Income

Invest P r t I Prt
Blue x 0.12 1 0.12x
Tech y 0.08 1 0.08y
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Example ? Simple Interest

• Translate TOTAL INTEREST statement

• Translate TOTAL Principal statement
• BluChip Prin. plus Tech Prin. is 10k

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Example ? Simple Interest

• Carry Out Solve Principal Eqn for y and then sub
  into Interest Eqn

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Example ? Simple Interest

• Back Substitute x 6500 into Principal Eqn to
  find y

?

• Check

?
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Example ? Simple Interest

• State Ms. Jeung invested3500 in technology
  Bonds and 6500 in blue-chip Bonds.

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WhiteBoard Work

• Problems From 3.2 Exercise Set
• 22

• PeppermintPattyPuzzled

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All Done for Today
CommercialCoffee BeanBlending
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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu