Transcript 8 tenets for wafer Problem Solving application

Problem Solving
Using the Eight Tenets
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High School
Technology
Initiative
© 2001
Introduction
• The eight tenet method of problem solving
lends itself well to mathematical solutions but
can be expanded to other processes.
• It uses a systematic approach to arrive at the
solution of a problem.
• This example revisits the problem solved in
the video and examines the solution in
greater depth.
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High School
Technology
Initiative
© 2001
The Eight Tenets of Problem Solving
1 Read and understand the
problem statement.
2 Draw and label a picture
that describes the problem
statement.
3 Determine the known and
unknown variables.
4 Examine the units and
convert all units to those
of the answer.
5 Determine the equations
to be used.
6 Solve the equations.
7 Check the physical
significance of the
answer.
8 Report the answer with
the correct units.
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High School
Technology
Initiative
© 2001
Tenet 1:
Read and Understand the Problem Statement
• The video question was :
• How many ten millimeter square chips can
fit on a circular wafer that has a diameter
of eight inches?
• This is a simple problem and you can think
of the square chips as squares and the
wafer as a circle.
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Technology
Initiative
© 2001
Tenet 2:
Draw and Label a Picture that Describes Problem
• The purpose of this tenet is as a visual aid for solving the
problem.
• Usually if you can picture the problem the solution is
easier to achieve.
A 10.0 mm by 10.0 mm Square
An Eight Inch Circle
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© 2001
Tenet 3:
Determine the Known and Unknown Variables
Known Variables :
• Diameter of Circle
– Dcir = 8.00 inches
• Length of a Side of a
Square
– Lside = 10.0 mm.
Unknown Variables :
• Radius of the Circle
• Area of the Circle
• Area of a Square
• Number of Squares
that fit inside the
circle.
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© 2001
Tenet 4:
Examine the Units Used in the Problem,
• Converting all of the units to the answer’s units
will save time in the end.
• The units of the circle are inches.
• The units of the sides of the square chip are
given millimeters.
• To solve this problem we need to convert the
units of the problem to millimeters.
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Tenet 4:
Examine the Units Used in the Problem.
• The units of the circle are expressed in inches and
must be converted to millimeters.
• The conversion factor from inches to centimeters is
2.54 centimeters per inch.
• There are 10 millimeters per centimeter.
• The units of the squares are correct as millimeters.
Diameter of an 8 inch circle in millimeters =
Diameter of a circle in millimeters =
 2.54 centimeters  10 millimeters 
(Diameter of a circle in inches) 


inch

 1 centimeter 
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High School
Technology
Initiative
© 2001
Tenet 4:
Examine the Units Used in the Problem.
Diameter
circle
in millimeters
=
Diameterof
ofaan
8 inch
circle in millimeters
=
 2.54 centimeters  10 millimeters 
(Diameter of a circle in inches) 


inch

 1 centimeter 
Using the fencepost method the units can
easily be converted from inches to millimeters
8.00 inches
2.54 cm
1 inch
10 mm
1 cm
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High School
Technology
Initiative
© 2001
Tenet 4:
Examine the Units Used in the Problem.
• How many ten millimeter square chips can
fit on a circular wafer that has a diameter
of eight inches?
Using the fencepost method the units can
easily be converted from inches to millimeters
8.00 inches
2.54 cm
1 inch
10 mm
1 cm
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High School
Technology
Initiative
© 2001
Tenet 4:
Examine the Units Used in the Problem.
• How many ten millimeter square chips can
fit on a circular wafer that has a diameter
of eight inches?
8.00 inches
2.54 cm
1 inch
10 mm
1 cm
=
(8) (2.54)
mm
(1) (1)
= 203.4 millimeters
Note;
the units cancel to give final length has units of millimeters.
At this point in the problem the calculation contains one
digit more than the correct number of significant figures.
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Tenet 5:
Determine the Equations to be Used
• To solve for the number of 10 mm by 10 mm squares
that will fit in an eight inch circle one must first solve
for the areas of the square and the circle,
• and then use these areas to solve for the number of
squares that will fit in the circle.
Area of one Square = Length of one side of a Square 
2
Area of the Circle = Radius 
2
Number of Squares = Area of the Circle
Area of one Square
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Tenet 5:
Determine Equations to be used.
Circle
• Radius of a circle from circle diameter
Rcircle = 1/2 (Dcircle)
• Area of a circle equation.
 Acircle =  (Rcircle)2
Square
• Area of a square.
 Asquare = (Lsquare)2
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Tenet 6:
Solving the Equations
Solving for the Areas of the Circle and a Square
R circle =
1
2

Dcircle  =
1
2
 203.2 mm 
A circle =  (R circle )2 = 3.14 101.6 mm 
= 101.6 mm
101.6 mm 
A circle = 32,410 mm2
A square = L square L square  = 10.0 mm
 10.0 mm 
A square = 100.0 mm2
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Tenet 6:
Solve the equations.
A circle = 32,410 mm2
A square = 100.0 mm2
A circle
32,410 mm2
Number of Squares =
=
2
A square
100.0 mm
chips
Number of Squares = 324.1
wafer
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Tenet 7:
Check the Units and Physical Significance of Answer
• Is 324.1 squares in that circle a suitable
answer?
• Does the answer make sense?
• Is the answer physically possible answer?
• The answer to the above three questions is
yes and no. To arrive at the solution we
rounded values and cut corners, literally.
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© 2001
Tenet 7:
Check units and physical significance.
Total are of square is 32,410 square millimeters.
First, overlay the picture
of the 324 Squares with
areas of 100 mm2 and the
eight inch circle with an
area of 32,410 mm2!
The two shapes have
equal area.
Therefore it is a good
mathematical solution.
Does it make physical sense.
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© 2001
Tenet 7:
Check units and physical significance.
Unfortunately it does not!
In semiconductor manufacturing only the complete
chips have a possibility of working. Therefore any
partial chip must be discarded.
Counting the number of
complete chips in the to
scale diagram to the left
yields a result of 289
complete squares to the
eight inch diameter circle.
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Tenet 8:
Report the Final Answer
• The unit for the number of squares is mm2 squares
per 8.00 inch diameter circle.
• When the area of the 8.00 inch diameter circle was
matched with the area of the 100 mm2 squares
number of squares the result was 324.
• The final answer (with the proper units).
There are less than 324 one hundred square
millimeter devices per 8.00 inch diameter
circular wafer.
Notice that now there are three significant
figures used for reporting the final answer!
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Tenet 8:
Report the final answer.
If an exact answer is desired a graphical solution
can be employed.
The exact answer, solved using a graphic method, is 289 one
hundred square millimeter chips per eight inch circular wafer.
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Problem Solving Using the Eight Tenets
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High School
Technology
Initiative
© 2001