4.5 Optimization II

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4.5 Optimization II

• Dr. Julia Arnold
• using Tans 5th edition Applied Calculus for the
  managerial , life, and social sciences text

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Suppose you want to make a rectangular garden but
you can only afford to buy 50 feet of fencing.
What would be the largest possible rectangle that
you could have.
This problem is different from the ones in the
last section in that you want to optimize one
thing (find max or min) but you have been given a
constraint (limited amount of fencing in this
case).
We need to consider two formulas one for the
perimeter of the rectangle (which represents the
fencing) and one for the area of the rectangle
(which represents the largest size garden.
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Suppose you want to make a rectangular garden but
you can only afford to buy 50 feet of fencing.
What would be the largest possible rectangle that
you could have.
We need to consider two formulas one for the
perimeter of the rectangle (which represents the
fencing) and one for the area of the rectangle
(which represents the largest size garden.
This problem is different from the ones in the
last section in that you want to optimize one
thing (find max or min) but you have been given a
constraint (limited amount of fencing in this
case).
The formula for the area which we would like to
be a maximum is A lw The formula for the
perimeter which is the constraint is 50 2l 2w
Since the Area formula is the one for which we
seek a maximum, it is the one that we need to
find the derivative. But, it has two variables l
and w. Thats a problem.
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Suppose you want to make a rectangular garden but
you can only afford to buy 50 feet of fencing.
What would be the largest possible rectangle that
you could have.
The formula for the area which we would like to
be a maximum is A lw The formula for the
perimeter which is the constraint is 50 2l 2w
A l w
Now substitute for l in
A l w A(w) (25-w)w
Now Area is only in terms of one variable w and
we can differentiate.
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The formula to maximize
The constraint equation
A l w A(w) (25-w)w
50 -2w 2l 25 - w l
Lets find A(w) A(w) (25 - w)1 w(-1)
product rule Simplify A(w) 25 - w - w 25 -
2w Set 0 0 25 - 2w 2w 25 w 12.5
To find l go back to the constraint equation and
substitute the w you just found.
50 -2w 2l 25 - w l 25 - 12.5 12.5 l
What kind of rectangle has l 12.5 and w 12.5?
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A rectangle that is a square.
12.5
The perimeter is 12.5 (4) 50 ft. of
fencing. Area is 156.25 sq. ft. or ft2
12.5 ft.
12.5
How do we know this is the maximum area?
12.5 ft.
A quick way is to look at the graph of our
function A(w) 25w - w2
At x 12.5 we reach the peak of the curve or the
vertex.
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Another way to convince ourselves that we have
the maximum area is to compute the area of some
other rectangles with perimeters of 50.
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Guidelines for Solving Optimization Problems

• Assign a letter to each variable mentioned in the
  problem. If appropriate, draw and label a
  figure.
• Find an expression for the quantity to be
  optimized.
• Use the conditions given in the problem (the
  constraint) to write the quantity to be optimized
  as a function f of one variable. Note any
  restrictions to be placed on the domain of f from
  physical considerations of the problem
• Optimize the function f over its domain using the
  methods of Section 4.4

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Example 1 Parcel Post Regulations Postal
regulations specify that a parcel sent by parcel
post may have a combined length and girth of no
more than 108 in. Find the dimensions of the
cylindrical package of greatest volume that may
be sent through the mail. What is the volume of
such a package? Hint The length plus the girth
is and the volume is
First draw (as best you can) a picture of the
cylindrical package.
Next, label the picture r is the radius of the
circular ends and l is the length of the cylinder.
r
In this problem the formulas are given in the
hint. Can you write the equation to be maximized?
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Example 1 Parcel Post Regulations Postal
regulations specify that a parcel sent by parcel
post may have a combined length and girth of no
more than 108 in. Find the dimensions of the
cylindrical package of greatest volume that may
be sent through the mail. What is the volume of
such a package? Hint The length plus the girth
is and the volume is
In this problem the formulas are given in the
hint. Can you write the equation to be maximized?
If you said you would be
correct as it says, Find the dimensions of the
cylindrical package of greatest volume and V
would stand for volume.
r
How many variables are in the Volume formula?
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Example 1 Parcel Post Regulations Postal
regulations specify that a parcel sent by parcel
post may have a combined length and girth of no
more than 108 in. Find the dimensions of the
cylindrical package of greatest volume that may
be sent through the mail. What is the volume of
such a package? Hint The length plus the girth
is and the volume is
How many variables are in the Volume formula?
Two r and l.
If there are two what did you learn from the
earlier problem about the garden?
r
Two variables is a problem. But the constraint
equation can fix it.
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Example 1 Parcel Post Regulations Postal
regulations specify that a parcel sent by parcel
post may have a combined length and girth of no
more than 108 in. Find the dimensions of the
cylindrical package of greatest volume that may
be sent through the mail. What is the volume of
such a package? Hint The length plus the girth
is and the volume is
What is the constraint equation?
Which is the easiest variable to solve for r or l
?
r
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The equation to maximize.
The constraint equation
The critical points are 0 and
Finding the derivative of V( r)
0 creates a minimum because the radius of 0 would
mean no cylinder. How do we know that
creates a maximum?
Lets use the 2nd derivative test.
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Since the 2nd derivative is negative that means
the function is concave down which means the
critical point is creating a maximum.
Example 1 Parcel Post Regulations Postal
regulations specify that a parcel sent by parcel
post may have a combined length and girth of no
more than 108 in. Find the dimensions of the
cylindrical package of greatest volume that may
be sent through the mail. What is the volume of
such a package? Hint The length plus the girth
is and the volume is
We have found the dimension of the radius to be
Is the length of the package.
The volume is
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Example 2 By cutting away identical squares
from each corner of a rectangular piece of
cardboard and folding up the resulting flaps, the
cardboard may be turned into an open box. If the
cardboard is 16 inches long and 10 inches wide,
find the dimensions of the box that will yield
the maximum volume.
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Can you figure out the inside dimensions?
Hint The long one is 16 2x.
The short one is 10 2x.
We need this information because they form the
dimensions of the box when you cut out the
corners and bend up the sides.
x
10-2x
16-2x
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We are constrained by the size of the paper we
are using. The volume of a rectangular solid
is V LWH (length, width, height) V(x)
(16-2x)(10-2x)x
To maximize the volume we take v(x).
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Now we solve for the value x has to be
In order to determine which of these numbers
gives a max or a min we can use the second
derivative test.
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Thus when x 2 we get the box with maximum
volume.
Just for fun lets compute a few volumes for
different x values.
As you can see x2 produces the largest volume.
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Minimizing Cost Example
For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
To get some information on Volume see the next
slide
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A cylinder is what we might think of as a can.
While we may have in mathematics slanted cans,
the ones in the store are what we call a right
circular cylinder in that the sides are
perpendicular to the horizontal. The base and
top of the can is a circle and thus has a radius
r, the distance between the top and bottom is
called the height of the can or h. If cut and
straightened out this shape would be a rectangle.
Click for sound
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For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
What is the constraint?
The volume.
The material which makes up the can.
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For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
What is the constraint?
If you chose
The volume.
Right, 36in3 to be precise.
The material which makes up the can.
No, this is what we want to minimize.
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For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
Write a formula for the volume and,
Write a formula for the material which makes up
the can.
Wait to click till you do.
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For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
Write a formula for the volume and,
Write a formula for the material which makes up
the can.
Top bottom sides
Which totals
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Explanation of formulas
Is the area of the circle. Multiply that by the
height to get Volume.
The can has a top and bottom whose area is a
circle thus
The side of the can rolls out into a rectangle in
which the width is h and the length is the same
as the Circumference of the Circle which is
. The area of a rectangle is LW
Added together
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The Volume constraint is written
The equation to minimize is which has two
variables r and h and as such cannot yet have
its derivative taken.
Next we need to substitute our value for h in the
constraint equation into the f(r,h) which would
then make it just f(r ).
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To minimize we find f(r ) and set equal to 0.
To check that this value does create a minimum,
lets do the 2nd derivative test.
Which is positive and thus creates a minimum.
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For its beef stew, the Betty Moore Company uses
tin containers that have the form of a right
circular cylinder (or a can). Find the radius and
the height of a container if it has a capacity of
36in3 and is constructed using the least amount
of metal.
We also need to find the height of the can with
this radius.
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This is what we have found The can has a
capacity of 36 in3 The radius and height which
will give the minimum amount of material used is
First, lets check that the volume is 36 in3.
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Now lets check to see if the material used is a
minimum amount by choosing values for r around
the value we obtained.
36
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Good Luck on the problems!