Section 44a To optimize something means to maximize or minimize some aspect of it Strategy for Solving MaxMin Problems 1 Understand the Problem Read the problem carefully Identify the information you need to solve the problem ID: 213712
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Slide1
Modeling and Optimization
Section 4.4aSlide2
To
optimize
something means to maximize or minimizesome aspect of it…
Strategy for Solving Max-Min Problems
1. Understand the Problem. Read the problem carefully. Identify the information you need to solve the problem.
2. Develop a Mathematical Model of the Problem. Draw pictures and label the parts that are important to the problem. Introduce a variable to represent the quantity to be maximized minimized. Using that variable, write a function whose extreme value gives the information sought.
3. Graph the Function.
Find the domain of the function.
Determine what values of the variable make sense in the
problem.Slide3
To
optimize
something means to maximize or minimizesome aspect of it…
Strategy for Solving Max-Min Problems
4. Identify the Critical Points and Endpoints. Find where
the derivative is zero or fails to exist.5. Solve the Mathematical Model. If unsure of the result, support or confirm your solution with another method.
6. Interpret the Solution.
Translate your mathematical
result into
the problem setting and decide whether the result
makes sense
.Slide4
Guided Practice
An open-top box is to be made by cutting congruent squares of
side length
x
from the corners of a 20- by 25-inch sheet of tin and
bending up the sides. How large should the squares be to make
the box hold as much as possible? What is the resulting max.
volume?
First, sketch
a diagram…
Dimensions of the box?
by
by
Volume of the box?
Domain restriction?Slide5
Guided Practice
An open-top box is to be made by cutting congruent squares of
side length
x
from the corners of a 20- by 25-inch sheet of tin and
bending up the sides. How large should the squares be to make
the box hold as much as possible? What is the resulting max.
volume?
Graphical solution?
Graph
,
calc. the maximum
Analytic solution?
CP
:
Check Endpoints and Critical Points:Slide6
Guided Practice
An open-top box is to be made by cutting congruent squares of
side length
x
from the corners of a 20- by 25-inch sheet of tin and
bending up the sides. How large should the squares be to make
the box hold as much as possible? What is the resulting max.
volume?
Interpretation?
Cutting out squares that are roughly 3.681 inches on a
side yields
a maximum volume
of approximately
820.528
cubic inches
.Slide7
Guided Practice
You have been asked to design a one-liter oil can shaped like
a right circular cylinder. What dimensions will use the least
material?
Sketch a
diagram…
Volume (in cm ):
Surface Area (in cm ):
2
3
We need to find a radius and height that will minimize the
surface area, all while satisfying the constraint given by
the volume…Slide8
Guided Practice
You have been asked to design a one-liter oil can shaped like
a right circular cylinder. What dimensions will use the least
material?Slide9
Guided Practice
You have been asked to design a one-liter oil can shaped like
a right circular cylinder. What dimensions will use the least
material?
A
is differentiable, with
r
> 0, on an interval with no endpoints.
Check the first derivative:
So, what happens
at this
r-value
?Slide10
Guided Practice
You have been asked to design a one-liter oil can shaped like
a right circular cylinder. What dimensions will use the least
material?
Check the second derivative:
This function is positive for all
positive
r
-values
, so the graph
of
A
is concave up for the entire
domain…
Which means our
r
-value
signifies an abs. min.Slide11
Guided Practice
You have been asked to design a one-liter oil can shaped like
a right circular cylinder. What dimensions will use the least
material?
Solve for height:
Interpretation:
The one-liter can that uses the least material has
height equal
to the diameter, with a radius of
approximately 5.419
cm and a height of approximately 10.839 cm.Slide12
Guided Practice
A rectangle has its base on the
x
-axis and its upper two vertices
on the parabola . What is the largest area the
rectangle can have, and what are its dimensions?
Dimensions of the rectangle:
by
Domain:
Model the situation:
Area of the rectangle:Slide13
Guided Practice
A rectangle has its base on the
x
-axis and its upper two vertices
on the parabola . What is the largest area the
rectangle can have, and what are its dimensions?
CP:
This corresponds to
a
maximum area (…
Why?
)
Largest possible area is A(2) =
32,
dimensions
are 4 by 8.
Maximize analytically:
Model the situation:Slide14
Guided Practice
A 216-m rectangular pea patch is to be enclosed by a fence
and divided into two equal parts by another fence parallel to
one of the sides. What dimensions for the outer rectangle will
require the smallest total length of fence? How much fence
will be needed?
2
Model the situation:
Area of the patch:
Total length of the fence:Slide15
Guided Practice
A 216-m rectangular pea patch is to be enclosed by a fence
and divided into two equal parts by another fence parallel to
one of the sides. What dimensions for the outer rectangle will
require the smallest total length of fence? How much fence
will be needed?
2
Maximize analytically:
CP:
This corresponds to
a
m
inimum length (…
Why?
)
The pea patch
will measure
12 m by 18
m (with
a
12-m divider
), and the total amount of fence needed is 72 m.