Transcript Problem Solving Methods

Team Exercise
 If you have enough money to buy a car,
what kind of car do you like to buy?
 If you are a car design engineer, identify
design goal and design parameters from
your team’s preference
 Taken from - http://homepages.stmartin.edu/
ETP 2005 – Dan Houston
This material is based upon work supported by the National Science Foundation
under Grant No. 0402616. Any opinions, findings and conclusions or
recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the view of the National Science Foundation (NSF).
Team Exercise
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Well Posed Design Problem: Design a
new car that can:
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1. Go from 0 - 60 mph in 6 seconds
2. Gets 50 miles/gal
3. Costs less than $10,000 to the consumer
4. Does not exceed government pollution
standards
5. Appeals to aesthetic tastes
Team Exercise
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1. Identify Problem e.g. we need to
build a new car since we are losing
market share
2. Synthesis (integrating parts to for a
whole) e.g. we can combine an
aerodynamic body with a fuel efficient
engine to make a new car with very
high fuel efficiency
Team Exercise
3. Analysis
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identify relationships,
distinguish fact from opinion,
detect logic information,
make conclusions from evidence,
select relevant information,
TRANSLATE REAL-WORLD PROBLEM INTO
MATHEMATICAL MODEL
e.g. compare the drag of different body
types and determine if engine can fit under
the hood
Team Exercise
4. Application (identify the pertinent
information) e.g. What force is required
to allow the car to go 60 mph knowing
the car has a 30ft2 projected area and a
0.35 drag coefficient based on wind
tunnel data?
Team Exercise
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5. Comprehension (use the data and
explicit theory to solve the problem)
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F = 1/2 Cd  A V2
F=force
Cd=drag coef. =air density A=protected
frontal area V=speed
Difficulties in Problem Solving
 Most common difficulty: failure to use known
information.
 To avoid this problem:
 Write the problem in primitive form and
sketch an accurate picture of the setup (where
applicable).
 Transform the primitive statements to simpler
language.
 Translate verbal problems to more abstract
mathematical statement(s) and figures,
diagrams, charts, etc.
General Problem Solving
Method
Define and understand problem
1. Sketch the problem
2. Gather information
3. Generate and evaluate potential
solutions
 Use applicable theories and assumptions
4. Refine and implement solution
5. Verify and test solution
Define and Understand
 Understand what is being asked
 Describe input/output (I/O)
 what are you given
 knowns
 what are you trying to find
 unknowns
 Sketch the problem
Gather Information
 Collect necessary data
 List relevant equations/theories
 State all assumptions
Generate Solution Methods
 Apply theories and assumptions.
 Typically, there is more than one approach
to solving a problem
 Work problem by hand using the potential
solution methods
 Break problem into parts; scale it down; etc.
 e.g., if the problem was to calculate the average
of 1000 numbers, work the problem by hand
using, say, 10 numbers, in order to establish a
method
Refine and Implement
 Evaluate solution methods.
 accuracy
 ease of implementation
 etc.
 Implement “best” solution.
Verify and Test
 Compare solution to the problem statement
 Is this what you were looking for?
 Does your answer make sense?
 Clearly identify the solution
 Sketch if appropriate
CHECK YOUR WORK!!
 Don’t stop at getting an answer!!
 Think about whether the answer makes
physical sense.
 you are the instructor and you have to turn in
final grades. In your haste, you calculate the
average of Susie’s grades (100, 70, 90) to be
78 and give Susie a C...
Getting It Right
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The problem solving process may be an
iterative process.
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If at first you don’t succeed (i.e., the
algorithm test fails), try again…
The more thorough you are at each
step of the problem solving process, the
more likely you are to get it right the
first time!!
Team Exercise
 Given: A student is in a stationary hotair balloon that is momentarily fixed at
1325 ft above a piece of land. This pilot
looks down 60o (from horizontal) and
turns laterally 360o.
Note: 1 acre = 43,560 ft2
Team Exercise; cont’
 Required:
 a) Sketch the problem
 b) How many acres of land are
contained by the cone created by her
line of site?
 c) How high would the balloon be if,
using the same procedure, an area
four times greater is encompassed?
Creative Problem Solving
 The nine dots shown
are arranged in equally
spaced rows and
columns. Connect all
nine points with four
straight lines without
lifting the pencil from
the paper and without
retracing any line.
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Individual Exercise (3 minutes)
Creative Problem Solving
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Creative Problem Solving
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If you enjoy solving puzzles, you will enjoy
engineering
Crick and Watson figured DNA when they
were young
Engineers create from nature what did not
exist before
In this creative process, the engineer
marshals skills in mathematics, materials, and
other engineering discipline and from these
resources create a new solution for a human
need
Creative Problem Solving
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Engineering is not dull or stifling; send
people to moon, communication from
battlefield, etc
Creative artists spent many years
perfecting their skills
Engineers need patience, practice, and
gaining problem-solving techniques by
training
Self-Questions for Problem Solving
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How important is the answer to a given
problem?
Would a rough, preliminary estimate be
satisfactory or high degree accuracy
demanded?
How much time do you have and what
resources are at your disposal?
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Data available or should be collected,
equipments and personnel, etc
Self-Questions for Problem Solving
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What about the theory you intend to use?
Can you use it now or must learn to use it?
Is it state of the art?
Can you make assumptions that simplify
without sacrificing needed accuracy?
Are other assumptions valid and applicable?
Optimize time and resources vs reliability
Engineering Method
1.
2.
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Recognize and understand the
problem (most difficult part)
Accumulate data and verify accuracy
Select the appropriate theory or
principles
Make necessary assumptions
Solve the problem
Verify and check results
Engineering Method
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Perfect solutions to real problems do
not exist. Simplify the problem to solve
it; steady state, rigid body, adiabatic,
isentropic, static etc
To solve a problem, use mathematical
model; direct methods, trial-and-error,
graphic methods, etc.
Problem Presentation
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Problem statement
Diagram
Theory
Assumptions
Solution steps
Identify results and verify accuracy
Standards of Problem Presentation
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Engineers should have ability to present
information with great clarity in a neat,
careful manner
Poor engineering documents can be
legal problems in courts
Follow standard forms such as shown in
the textbooks
Team Assignment
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Page 141 Problem 3.20
Algorithms
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Algorithm: “a step-by-step procedure
for solving a problem or accomplishing
an end” (Webster)
Algorithms can be described by
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Pseudocode
Flowcharts
Pseudocode
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English-like description of each step of
algorithm
Not computer code
Example - take out trash barrels
while there are more barrels
take barrel to street
return to garage
end
Flowcharts
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Graphical description of algorithm
Standard symbols used for specific
operations
Input/Output
Start/Stop
Branch Test
Process Step
Process Flow
Flowchart Example
Define the
problem
Read
input
Begin
Ask for
more input
yes
Can I
solve this?
yes
Solve the
problem
no What do I need
to know?
Can I
solve this?
no
Output
results
End
Top Down Design
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State problem clearly
Sketch problem
Describe input/output (I/O)
Work problem by hand
Algorithm: pseudocode or flowchart
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Decomposition - break problem into steps
Stepwise refinement - solve each step
Test the algorithm/check your work!!
Example (Team exercise, 15
min)
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State problem clearly:
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Given ax2 + bx + c = 0, find x.
Describe I/O:
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Input: a, b, c
Output: x
Example (cont.)
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Hand example:
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a=1, b=4, c=4
equation? (See Chapter 6, Mathematics
Supplement)
x=?
Example (cont.)
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Algorithm development
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write an algorithm in pseudocode to take
any set of coefficients (i.e., a, b, c) and
give the value of x for each set
Test your algorithm
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a,b,c = 1,4,4
a,b,c = 1,1,-6
a,b,c = 1,0,1
other good test cases?