Modeling with Linear Equations by yurtgc548

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									 Modeling with Linear
     Equations




Introduction to Problem Solving
    Ex 1 Using a Verbal Model
• You have accepted a job for which your annual salary
  will be $32,300. This salary includes a year-end
  bonus of $500. You will be paid twice a month.
  What will your gross pay (before taxes) be for each
  paycheck?
                      Ex. 1
• Your income ($32,300) will be the 24 paychecks plus
  your $500 bonus.
• The unknown is the amount of each paycheck. Let’s
  make that x.
                      Ex. 1
• Your income ($32,300) will be the 24 paychecks plus
  your $500 bonus.
• The unknown is the amount of each paycheck. Let’s
  make that x.
• Your pay can be expressed by the equation:
           32,300 = 24x + 500
                      Ex. 1
• Your income ($32,300) will be the 24 paychecks plus
  your $500 bonus.
• The unknown is the amount of each paycheck. Let’s
  make that x.
• Your pay can be expressed by the equation:
           32,300 = 24x + 500
           31,800 = 24x
           $1,325 = x
    Ex. 2 Finding the Percent of a
                Raise
• You have accepted a job that pays $8 an hour. You
  are told that after a two-month probationary period,
  your hourly wage will be increased to $9 an hour.
  What percent raise will you receive after the two-
  month period.
                       Ex. 2
•   What do we know?
•   Original amount = 8
•   New amount      =9
•   Amount of change = 1
                         Ex. 2
•   What do we know?
•   Original amount = 8
•   New amount       =9
•   Amount of change = 1
•   The equation we will use is:

    amount of change = % change
     original amount
                         Ex. 2
•   What do we know?
•   Original amount = 8
•   New amount       =9
•   Amount of change = 1
•   The equation we will use is:

    amount of change = % change
     original amount
    1
       %change         .125  %change   12.5%
    8
     Ex. 3 Finding the Percent of
          Monthly Expenses
• Your family has an annual income of $57,000 and the
  total monthly expenses of $26,760. Monthly
  expenses represents what percent of your family’s
  annual income?
                      Ex. 3
• Your family has an annual income of $57,000 and the
  total monthly expenses of $26,760. Monthly
  expenses represents what percent of your family’s
  annual income?
• The equation we will use here is:
         part = %
         whole
                      Ex. 3
• Your family has an annual income of $57,000 and the
  total monthly expenses of $26,760. Monthly
  expenses represents what percent of your family’s
  annual income?
• The equation we will use here is:
         part = %      26,760 = .469 or 46.9%
         whole         57,000
 Ex. 4 Finding the Dimensions of a
               Room
• A rectangular kitchen is twice as long as it is wide,
  and its perimeter is 84 feet. Find the dimensions of
  the kitchen.
 Ex. 4 Finding the Dimensions of a
               Room
• A rectangular kitchen is twice as long as it is wide,
  and its perimeter is 84 feet. Find the dimensions of
  the kitchen.
• We always want to make x the thing that we are
  comparing other things to. In this problem, we are
  comparing the length to the width.
                       Ex. 4
• This means that the width is x.
• The length is twice the width, so that means that
  Length = 2(width) or 2x.
                       Ex. 4
• This means that the width is x.
• The length is twice the width, so that means that
  Length = 2(width) or 2x.
• The perimeter of a rectangle is:
    2(length) + 2(width) = P
                       Ex. 4
• This means that the width is x.
• The length is twice the width, so that means that
  Length = 2(width) or 2x.
• The perimeter of a rectangle is:
    2(length) + 2(width) = P
    2(2x)    + 2x       =P
       6x=84, x = 14, so:
     w = 14ft, and l = 28ft
      Ex. 5 A Distance Problem
• A plane is flying nonstop from Atlanta to Portland, a
  distance of about 2700 miles. After 1.5 hrs. in the
  air, the plane flies over Kansas City (a distance of 820
  miles from Atlanta). Estimate the time it will take
  the plane to fly from Atlanta to Portland.
                       Ex. 5
• The equations we will use are:
        Rate x Time = Distance

   or       Distance = Time
              rate

   or        Distance = rate
              time
                           Ex. 5
• We know that it took 1.5 hrs. to fly 820 miles, so to
  find the rate, we use:

          Distance = rate
            time


         820
              546.66m / h
         1.5
                      Ex. 5
• We now know the rate that the plane is flying. To
  find the time we use the equation:
               distance = time
                 rate


               2700
                      4.94hrs.
              546.66
        Ex. 6 Similar Triangles
• To determine the height of the Aon Center Building,
  you must use similar triangles. We know that a 4 ft.
  post will cast a shadow of 6 in. The Building will cast
  a shadow of 142 ft. Set up a proportion to solve for
  the height of the building.
                      Ex. 6
• Remember, in similar triangles, the corresponding
  sides are in proportion.

  Height of building = Shadow of Building
   Height of Post      Shadow of Post
                      Ex. 6
• Remember, in similar triangles, the corresponding
  sides are in proportion.

  Height of building = Shadow of Building
   Height of Post        Shadow of Post
• All units must be the same!!

       x 142
                    x  1136 ft
       4 .5
             Literal Equations
• An equation that contains more than one variable is
  called a Literal Equation. Formulas are a great
  example of literal equations. Look at the formulas on
  page 103. Many you know, some you may not.
            Literal Equations
• Here are some common equations that you will use
  in many of your math classes.
            Literal Equations
• Here are some common equations that you will use
  in many of your math classes.
        Ex. 9 Using a Formula
• A cylindrical can has a volume of 200 cubic
  centimeters and a radius of 4 cm. Find the height of
  the can.
                       Ex. 9
• A cylindrical can has a volume of 200 cubic
  centimeters and a radius of 4 cm. Find the height of
  the can.
• The formula for the volume of a can is:



                   V  r h  2
                        Ex. 9
• Since we are trying to find the height, we will solve
  the equation for height first.

                                   V
     V  r h   2
                                h 2
                                  r
                        Ex. 9
• Since we are trying to find the height, we will solve
  the equation for height first.

                                   V
     V  r h   2
                                h 2
                                  r

        200
    h                       h  3.98cm.
        ( 4) 2
            Percent Problems
• There is another type of percent problem that is very
  basic. Some examples are:
• What is 30% of 70?
• 12 is 20% of what number?
• 112 is what % of 300?
            Percent Problems
• There is another type of percent problem that is very
  basic. Some examples are:
• What is 30% of 70?
• 12 is 20% of what number?
• 112 is what % of 300?
• The word what becomes the variable x.
• The word is now means equals.
• The word of now means multiplication.
            Percent Problems
• There is another type of percent problem that is very
  basic. Some examples are:
• What is 30% of 70?                 x  (.30)  (70)
• 12 is 20% of what number?          12  (.20)  ( x)
• 112 is what % of 300?             112  ( x)  (300)
• The word what becomes the variable x.
• The word is now means equals.
• The word of now means multiplication.
              Percent Problems
• There is another type of percent problem that is very
  basic. Some examples are:
• What is 30% of 70?
• 12 is 20% of what number?
• 112 is what % of 300?

 x  (.30)  (70)   12  (.20)  ( x)   112  ( x)  (300)
 x  21             12                  112
                                             x
                        x
                    .20                 300
                    x  60              x  .37or37%
            Class work
• Pages 105-106
• 12-22 even
               Homework
•   Pages 105-106
•   11-21 odd
•   35-53 odd
•   63

								
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