Optimization

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Optimization
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In optimization problems we are trying to find
the maximum or minimum value of a variable. The
solution is called the optimum solution.
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Optimization Problem Solving Method
Step 1 If possible, draw a large, clear diagram.
Sometimes more than one diagram is necessary
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Optimization Problem Solving Method
Step 2 Construct an equation with the variable
to be optimized (maximized or minimized) as the
subject of the formula(the y in your calculator)
in terms of one convenient variable, x. Find any
restrictions there may be on x.
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Optimization Problem Solving Method
Step 3 Find the first derivative and find the
value(s) of x when it is zero
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Optimization Problem Solving Method
Step 4 If there is a restricted domain such as
a?x?b, the maximum/minimum value of the function
may occur either when the derivative is zero or
at xa or at xb. Show by a sign diagram that you
have a maximum or minimum situation.
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A industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 1 If possible, draw a large, clear diagram.
Sometimes more than one diagram is necessary.
x m
y m
8
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 2 Construct an equation with the variable
to be optimized as the subject of the formula in
terms of one convenient variable, x. Find any
restrictions there may be on x.
What do we need to know?
COST
What is the formula for cost?
COST60(Total Length of the Walls)
9
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 2 Construct an equation with the variable
to be optimized as the subject of the formula in
terms of one convenient variable, x. Find any
restrictions there may be on x.
Therefore the formula for cost is?
C60(6x4y)
How do you find the total length of the walls?
L6x4y
10
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 2 Construct an equation with the variable
to be optimized as the subject of the formula in
terms of one convenient variable, x. Find any
restrictions there may be on x.
REMEMBER We want to have our formula in terms of
one variable!
So what else does the problem tell us?
What is the formula for area in terms of x and y?
Area600m2
3xy
11
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 2 Construct an equation with the variable
to be optimized as the subject of the formula in
terms of one convenient variable, x. Find any
restrictions there may be on x.
Area600m23xy
y200 x
Solve for y (to get y in terms of x)
Substitute for y in cost formula
C60(6x4(200)) x
12
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 2 Construct an equation with the variable
to be optimized as the subject of the formula in
terms of one convenient variable, x. Find any
restrictions there may be on x.
C60(6x4(200)) x
Do we have any restrictions on x or y?
YES - x?0 and y?0
13
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 3 Find the first derivative and find the
value(s) of x when it is zero
C60(6x4(200)) x
C360x48000x-1
C360-48000x-2360-48000 x2
0360-48000 x2
36048000 x2
360x248000
x2?133.3
x?11.547
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An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 4 If there is a restricted domain such as
a?x?b, the max/min value of the function may
occur either when the derivative is 0 or at xa
or at xb. Show by a sign diagram that you have a
max or min.
Put in calculator first!
What does the graph show you?
The end points are not going to be where cost is
minimized, and where the derivative0 is a min
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An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
Step 4 If there is a restricted domain such as
a?x?b, the max/min value of the function may
occur either when the derivative is 0 or at xa
or at xb. Show by a sign diagram that you have a
max or min.
What would the sign diagram look like?


x?11.547
Could see this by putting derivative in your
calculator and looking at the table
16
An industrial shed is to have a total floor space
of 600 m2 and is to be divided into 3 rectangular
rooms of equal size. The walls, internal and
external, will cost 60 per meter to build. What
dimensions would the shed have to minimize the
cost of the walls?
But the question still hasnt been answered
What are the dimensions?
3x meters by y m
3x meters by (200/x) m
3(11.547) meters by (200/11.547) m
34.6 meters by 17.3 m
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An open rectangular box has square base and a
fixed outer surface area of 108 cm2. What size
must the base be for maximum volume?
ANSWER 6 cm by 6 cm or 36 cm2
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Solve using derivatives Square corners are cut
from a piece of 20 cm by 42 cm cardboard which is
then bent into the form of an open box. What size
squares should be removed if the volume is to be
maximized?
YOU DO
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TICKET OUT

• A closed box has a square base of side x and
  height h.
• Write down an expression for the volume, V, of
  the box
• Write down an expression for the total surface
  area, A, of the box
• The volume of the box is 1000 cm3
• Express h is terms of x
• Write down the formula for total surface area in
  terms of x
• Find the derivative (dA/dx)
• Calculate the value of x that gives a minimum
  surface area
• Find the surface area for this value of x