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Problem Solving

Using the Eight Tenets

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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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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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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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Tenet 2 : Draw and Label a Picture that Describes the

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.

An Eight Inch Circle

A 10.0 mm * 10.0 mm Square

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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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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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• 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.

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

Tenet 4 : Examine the Units Used in the Problem

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8.00 inches

2.54 cm

inch

10 mm

cm

Tenet 4 : Examine the Units Used in the Problem

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

Using the fencepost method the units caneasily be converted from inches to millimeters

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Tenet 4 : Examine the Units Used in the Problem

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

8.00 inches

2.54 cm

inch

10 mm

cm

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Tenet 4 : Examine the Units Used in the Problem

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

8.00 inches

2.54 cm

inch

10 mm

cm

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= 203.2 mm

Notice that the units cancel and that the final length has units of millimeters.

Tenet 4 : Examine the Units Used in the Problem

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

8.00 inches

2.54 cm

inch

10 mm

cm

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= 203.2 mm

Also notice that one digit more than the number of significant figures is being carried through the problem.

Tenet 4 : Examine the Units Used in the Problem

1

Diameter of a circle in millimeters =

2.54 centimeters 10 millimeters(Diameter of a circle in inches)

inch centimeter

8.00 inches

2.54 cm

inch

10 mm

cm

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• 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.

2

2

Length of one side of a Square

Radius

Area of one Square =

Area of the Circle =

Area of the CircleNumber of Squares = Area of one Square

Tenet 5 : Determine the Equations to Be Used

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• Radius of a circle from diameter of a circle. Rcircle = (Dcircle)

• Area of a circle equation. Acircle = (Rcircle)2

• Area of a square. Asquare = (Lsquare)2

Tenet 5 : Determine the Equations to Be Used

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circle

circle

circle

2circle

2

square square

2

circle

square

square

1 1 = D = 203.2 mm = 101.6 mm

2 2

= (R ) = 3.14 101.6 mm 101.6 mm

= 32,410 mm

= L L = 10.0 mm

R

10.0 mm

= 100.

A

A

A

A 0 mm

Solving for the Areas of the Circle and a Square

Tenet 6 : Solving the Equations

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2

2

2circle

2square

circle

square

= 32,410 mm

= 100.0 mm

A 32,410 mm =

A

A

Number of Squares

Number of Squar

= A 100.0 mm

chips = 3e 24.1s

wafer

Tenet 6 : Solving the Equations

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• 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.

Tenet 7 : Checking the Units and Physical Significance of the Answer

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Here is an overlay picture of the 324 Squares withareas of 100 mm2 and the eight inch circle with an area of 32,410 mm2!

They have equal area and therefore it is a good solution, or is it?

Tenet 7 : Checking the Units and Physical Significance of the Answer

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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.

In semiconductor manufacturing only the complete chips have a possibility of working. Therefore any partial chip must be discarded.

Tenet 7 : Checking the Units and Physical Significance of the Answer

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• 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 value with units is less than 324 100 mm2 squares per 8.00 inch diameter circle.

Tenet 8 : Reporting the Final Answer

Notice that three significant figures are used for reporting the final answer!

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If an exact answer is desired a graphical solution can be employed.

The exact answer, solved for graphically is289 100 mm2 square chips per eight inchcircular wafer.

Tenet 8 : Reporting the Final Answer