Transcript Optimal Design Process

OPTIMAL DESIGN PROCESS
USING SYSTEMS of INEQUALITIES
Warm Up
Answer the following questions on your note sheet:
1.
2.
Think about a time when you worked with a team
to solve a problem. How did you work together?
How did you settle disagreements?
Find the slope and y-intercept of each line:
a)
b)
c)
y = 3x + 7
y = 9x – 1
3x + y = 4
Warm Up Solutions
1.
2.
Discuss in a think-pair-share or
share out loud with class.
Find the slope and y-intercept of each line:
a)
b)
c)
y = 3x + 7
y = 9x – 1
3x + y = 4
m = 3; (0,7)
m = 9; (0,-1)
m = -3; (0,4)
Agenda
Time
5 mins
10 mins
20 mins
Activity
Warm Up
Lockheed Martin Intro
Direct Instruction
15 - 30 mins Guided Practice
20 - 45 mins Independent Practice/Poster
5 mins
Exit Slip
Lockheed Martin

Builds
 Satellites
(communication, surveillance)
 Missile defense (THAAD)
 Space Vehicles (Mars Phoenix Lander, Hubble Space
Telescope)
 Fighter Jets (F-16, F-22, F-35)


Has approximately 123,000 employees
#1 in Federal Defense contracts ($35.8 billion)
Where are the engineers?



A large number (more than 50%) of engineers are
nearing retirement age.
Not very many engineers graduating from college
to fill these positions.
What’s the problem?
Challenge Problem

We need to build an electrical box that will be
important for a functioning missile. The rectangular
box with square base must have a length no greater
than 6 inches and the base side no greater than 4
inches. The box cannot have a size more than 10
linear inches (length + width + height). What are the
best dimensions to minimize the surface area?
y
x
y
Design Teams




Once awarded contracts, design teams are formed
to design parts for the equipment.
Teams can be made up of up to 30 engineers
(or more!)
Each satellite, missile, and space vehicle may be
made up of hundreds of smaller parts.
Each part needs to meet specifications based on
weather, terrain, space, environmental concerns, etc.
Sample Specifications





The SSU shall have a ground life of not less than 7 years, of which 2 years
is to be dedicated to spacecraft testing and no more than 4 years will
accrue while stored in a completed spacecraft, under the storage
environments.
The SSU weight shall be no greater than 20 pounds.
Note: SSU will be designed to minimize weight.
The SSU center of gravity shall be verified to an accuracy of ±0.5 inches in
each of the X, Y, and Z axes.
The resistance between electrically common circuits at the SSU external
connector pins shall be no greater than: 1 ohm maximum for ground and
telemetry signals.
The SSU differential signal skew between signals in the same signal group
shall be no more than 100 nsecs.
SSU = Satellite Switching Unit
Vocabulary


Feasible Region: A “walled off” region in the xyplane created by many different inequalities.
Optimization Equation: an equation for which you
plug in values to find the maximum or minimum
values.
Teacher Models

Let’s start with a standard problem:
 Find
the maximum and minimum values of z = 3x +
4y subject to the following constraints:
 x  2 y  14

3x  y  0
x  y  2

Steps to Solve
Put all inequalities in slope-intercept form.
2. Find the y-intercept b, and the slope m.
3. Plot the y-intercept on the y-axis and use the slope (rise/run) to
plot the next point. Connect the points to form a line.
1.
1. Solid
line for ≥ or ≤.
2. Dotted line for > or <.
4.
Shade the appropriate side of the line.
1. Above
for > or ≥.
2. Below for < or ≤.
Find the intersection points (as ordered pairs) of the lines in the
feasible set.
6. Test all points in the optimization equation and answer question.
5.
Solution
Solve each of the inequalities
for y so it is easy to find m & b.
1

y   x7

x

2
y

14

2


3 x  y  0   y  3x
x  y  2
y  x  2



Graph the inequalities on the
same set of axes, find the
intersection of the regions
Find the slope (m)
and the y-intercept
(b) for each
inequality
m = -½, b = 7
m = 3, b = 0
m = 1, b = -2
Solution Continued
Find the intersections
algebraically (set equations
equal to each other)
y = 3x & y = x - 2
3x = x - 2
2x = -2
x = -1
y = 3(-1) = -3
(-1, -3)
Set equations
equal to each
other
Subtract x from
both sides
Divide both
sides by 2
Substitute value
into equation
y = 3x & y = -1/2x + 7
3x = -1/2x + 7
7/2x = 7
Set equations
equal to each
other
Add 1/2x to
both sides
x=2
Multiply both
sides by the
reciprocal 2/7
y = 3(2) = 6
Substitute value
into equation
(2, 6)
Intersection
Point
(2,6)
(6,4)
Intersection
Point
(-1,-3)
y = x - 2 & y = -1/2x + 7
x - 2 = -1/2x + 7
Set equations
equal to each
other
3/2x - 2 = 7
Add 1/2x to
both sides
3/2x = 9
Add 2 to both
sides
x=6
Multiply both
sides by the
reciprocal 2/3
y=6–2=4
Substitute value
into equation
(6, 4)
Intersection Point
Solution Continued
Plug points into optimization equation and find the maximum
and minimum values.
Point
z = 3x + 4y
Value
(-1, -3)
3(-1) + 4(-3)
-15
(2, 6)
3(2) + 4(6)
30
(6, 4)
3(6) + 4(4)
34
Minimum
Maximum
We Do Together

Problem 2: Find the maximum value of z = 3x – 7y
subject to the following constraints:
x  y  2

y  x
x  0

You Do

Problem 3: Find the minimum value of z = y + 2x
subject to the following constraints:
x  2 y  8

y  x  4
y  0

You Do Again (Extra)

Problem 4 (extra): Find the maximum and minimum
values of z = 6x2 + 5y subject to the following
constraints:
2 x  3 y  12

 x  5 y  20
x  0

Independent Practice

We need to build an electrical box that will be important
for a functioning missile. The rectangular box with square
base must have a length no greater than 6 inches and the
base side no greater than 4 inches. The box cannot have
a size more than 10 linear inches (length + width +
height). What are the best dimensions to minimize the
surface area?
y  4

x  6
2 y  x  10

Optimization Equation
SA = 4xy +
2
2y
Poster

Your Job:
 Solve
the challenge problem.
 Create a poster with the following elements:
 Problem
statement
 Graph of feasible set
 Explanation of how you graphed the inequalities
 All work showing your arrival at your final answer
 Explanation of your final answer
Exit Slip

After finishing your poster, answer the following
questions on your worksheet:
1.
2.
In your own words, explain what you learned today.
(5-7 complete sentences)
What are you still confused about? If you understood
everything, what concepts did you understand best
and least? (3-5 complete sentences)