warm up 9/10/14 a) find all relative extrema using the 2 nd derivative test: b) find any points of...
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3.7 Modeling and Optimization
Buffalo Bill’s Ranch, North Platte, NebraskaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999
3.7 Optimization ProblemsOne of the most common applications of calculus involves the determination of minimum and maximum values. Consider how frequently you hear or read terms such as…
A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
x x
40 2x
40 2A x x
240 2A x x
40 4A x
0 40 4x
4 40x
10x 40 2l x
w x 10 ftw
20 ftl
There must be a local maximum here, since the endpoints are minimums.
A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
x x
40 2x
40 2A x x
240 2A x x
40 4A x
0 40 4x
4 40x
10x
10 40 2 10A
10 20A
2200 ftA40 2l x
w x 10 ftw
20 ftl
To find the maximum (or minimum) value of a function:
1 Write it in terms of one variable.
2 Find the first derivative and set it equal to zero.
3 Check the end points if necessary.
Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?
We can minimize the material by minimizing the area.
22 2A r rh area ofends
lateralarea
We need another equation that relates r and h:
2V r h
31 L 1000 cm21000 r h
2
1000h
r
22
10 02
02A r r
r
2 20002A r
r
2
20004A r
r
Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?
22 2A r rh area ofends
lateralarea
2V r h
31 L 1000 cm21000 r h
2
1000h
r
22
10 02
02A r r
r
2 20002A r
r
2
20004A r
r
2
20000 4 r
r
2
20004 r
r
32000 4 r
3500r
3500
r
5.42 cmr
2
1000
5.42h
10.83 cmh
If the end points could be the maximum or minimum, you have to check those also.
Notes:
If the function that you want to optimize has more than one variable, use substitution to rewrite the function.
If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check by using either the 1st or 2nd derivative test.
You Try:
Find two positive numbers such that the sum of the first number squared and the second is 27 and the product is a maximum.
3.7 Optimization Problems
A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?
2V x h Primary Equa n tio2 4 108S x xh Secondary Equ n atio
(use the second
Express as a
ary equation t
function of
o write
one
i
variabl
n terms )
e
. of V x
V
2 4 108x xh 24 108xh x
2108
4
xh
x
22 108
4
xV x
x
3
274
xV x x x
x
h
Solution
3.7 Optimization Problems
Feasible Domain (real-world)?
must be positivex2 is less than 108bA x
0 108x ?dV
dx
2327
4
dV x
dx 0
2327
4
x
23 108x 2 36x
6x
(6) ?V 3108 in
If the domain was a closed interval,
we would also have to check for
extrema at its endpoints.
V
3
274
xV x x
Diagram
2
2
d y
dx 6x 3 6
9 02
Is this the maximum volume?
xx
h
3.7 Optimization Problems
2V x h2108 (6) h
3h
6" 6" 3"x x
6, 108, Solve for x V h
xx
h
Diagram is not drawn to scale
3.7 Optimization Problems
Which points on the graph of24y x
are closest to the point (0,2)?
2 2
The quantity to be minimized is distance:
0 2d x y
Work
2 2
2 1 2 1d x x y y
3.7 Optimization Problems
2 2( 2)d x y Primary
Eq n
uatio
24 Secondary Equ i n at oy x
2 224 2 ( )xd x 2 22 (2 )x x
2 2 4(4 4 )x x x
4 23 4x x
Use the secondary
equation to write
the primary equation
in terms of one variable.
Diagram
3.7 Optimization Problems
Find the minima of4 23 4d x x
because d is smallest when the radicand is smallest,
we simply need the minima of
4 2( ) 3 4f x x x 3'( ) 4 6f x x x
22 (2 3)x x 00 orx 22 3 0x
3
2x
Table
3.7 Optimization Problems
3 5 3 5Min. @ , & ,
2 2 2 2
Max. @ (0,4)
Closest points are 3 5 3 5, & ,
2 2 2 2
3'( ) 4 6f x x x
Double check with the 2nd Derivative Test:
2''( ) 12 6f x x
3'' 12 0
2f
3'' 12 0
2f
'' 0 6 0f
3.7 Optimization Problems
3 5- ,
2 2
3 5,
2 2
Note: 0 yields a relative maximum, there is no absolute maximum since the domain is the entire real line.
Homework:
Day 1: Pg. 223 5-21 odd & 29
Day 2 Problems
Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least wire?
The quantity to
be minimized is
length. From the
diagram you can
see that varies
between 0 and 30.
x
Domain?
3.7 Optimization Problems
Primary Equa n tioW y z 2 2 212y x 2 2 2(30 ) 28z x
2 2(30 28)xz 2900 60 784x x
2 60 1684x x 2 144y x
Secondary Equations!
Write y and z in terms of x.
3.7 Optimization Problems
Primary EquationW y z
in (0,30)x2 2144 60 1684W x x x 1 1
2 22 2( 144) ( 60 1684)x x x
1 12 22 2
2 60( )
( 144) ( 60 1684)2
x x
x x x
dw
dx
1 12 22 21 1
( 144) (2 ) ( 60 1684) (2 60)2 2x x x x x
Use the secondary equations to write the
primary equation in terms of .x
3.7 Optimization Problems
2 2
( 30)
144 60 1684
dw x x
dx x x x
0
dw
dx
2 2
( 30)0
144 60 1684
x x
x x x
2 2
30
144 60 1684
x x
x x x
2 260 168 (304 144)x xx x x
2 2 2 2( 60 1684) (30 ) ( 144)x x x x x
It could be the proportion
from &$##!
4 3 2 2 260 1684 (900 60 )( 144)x x x x x x
3.7 Optimization Problems
4 3 2 2 3 4 260 1684 900 60 129600 8640 144x x x x x x x x
2 2 21684 900 129600 8640 144x x x x 2 21684 1044 8640 129600x x x 2640 8640 129,600 0x x
2320(2 27 405) 0x x 320(2 45)( 9) 0x x 9, 22.5x
2 2 2 2( 60 1684) (30 ) ( 144)x x x x x
Obviously, 320 is a common factor
Factorable
Put in Quadratic Form! Combine Like Terms
3.7 Optimization Problems
2 2(9) 9 144 9 60(9) 1684w 50
The wire should be staked 9 feet from the 12
:
foot pole.
Conclusion
3.7 Optimization Problems
4 ' of wire is to be used to form a square and/or a circle.
How much of the wire should be used for the square and
how much should be used for the circle to enclose the
maximum total area?
2 2
The quantity to
be maximized is
area.
.A x r SolutionPrimary Equa n tio
4 ' of wire is to be used to form a square and/or a circle.
How much of the wire should be used for the square and
how much should be used for the circle to enclose the
maximum total area? Re member that an extreme value
can also occur at the endpoints of an interval.
2 2 Primary EquationA x r
4 4 2 Secondary Equationx r 4 4
2
xr
2 2x
2(1 )x
22 2(1 )x
A x
2
2 4(1 )xx
2 2(1 )4x x
2 21(1 2 )4x x x
2 21( 4 8 )4xx x
21(4 ) 8 4x x
3.7 Optimization Problems
Feasible Domain? in [0,1] x
The perimeter of the square could be as little as
zero or as much as 4.
(8 2 ) 80
dA
dx
x
(0) 1.273A 4
( ) 0.5604
A
(1) 1A The maximum area occurs
when 0. That is, x
Diagram
You could use all or
none of the wire for the
square.
21(4 ) 8 4x x
8 2 8 0
8 48 2 8
8 2 4
x x
x x
when all the wire is
used for the circle.
You Try:
A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?
3.7 Optimization Problems
You must expect that real-life applications often
involve equations that are at least as complicated
as the primary equations seen in today's examples.
Remember, one of the main goals of this course is
to learn to use calculus to analyze equations that
initially seem formidable.
HW Day 2 MMM pgs 111-113