Transcript Optimization Problems

Optimization Problems
Finding maximum and minimum
Example Problem: Product

Find two nonnegative numbers whose sum
is 9 and so that the product of one
number and the square of the other
number is a maximum.
General Guidelines for Optimization
1.
Read each problem slowly and carefully.

2.
If appropriate, draw a sketch or diagram of the problem to be solved.

3.
6.
7.
This step is very important because it leads directly or indirectly to the creation of
mathematical equations.
Write down all equations which are related to your problem or diagram.
Clearly denote that equation which you are asked to maximize or
minimize.

5.
Pictures are a great help in organizing and sorting out your thoughts.
Define variables to be used and carefully label your picture or diagram
with these variables.

4.
Read the problem at least three times before trying to solve it. Sometimes words can be
ambiguous. It is imperative to know exactly what the problem is asking. If you misread the
problem or hurry through it, you have NO chance of solving it correctly.
Experience will show you that MOST optimization problems will begin with two equations.
One equation is a "constraint" equation and the other is the "optimization" equation. The
"constraint" equation is used to solve for one of the variables. This is then substituted into
the "optimization" equation before differentiation occurs.
Determine the domain of the function and set an appropriate window on
the calculator.
Use the maximum/minimum function to find the values.
Interpret your results and answer the question.
Example Problem: Product

Find two nonnegative numbers whose sum is 9 and so that
the product of one number and the square of the other
number is a maximum.


Quantities: two numbers, x and y
Optimization Equation:


Constraint Equation:



y=9-x, so Product=x(9-x)2
IMPORTANT: Optimization Equation should contain ONLY ONE
variable.
Appropriate Domain:


Two numbers whose sum is 9: x+y=9
Substitute the constraint equation into the Optimization
Equation:


Trying to maximize product of x and y2: Product=xy2
Nonnegative numbers: Domain: [0, 9]
Graph and solve:

Answer: 3 and 6, Product = (3)(62)=108
Example Problem: Capacity

A manufacturer wants
to design an open box
(a box with open top)
having a square base
and a surface area of
108 square inches.
What dimensions will
produce a box with
maximum volume?

Optimization Equation:


Constraint Equation:


V=x2h
x2+4xh=108
Equation to maximize:


V=x2[(108-x2)/(4x)]
V=27x – x3/4
Example Problem: Capacity
Practice Problem: Distance

Which points on the
graph of y=4-x2 are
closest to the point
(0,2)?
Optimization Equation:
d
 x  0
2
  y  2
2
Constraint Equation:
y  4  x2
Equation to minimize:
d
 x  0
2
  4  x2  2  x2   2  x2 
2
Domain:  ,  
Reasonable Window: 0, 2
Answer:  1.225, 2.5 
2
Practice Problem: Area

A rectangle is
bounded by x-axis and
the top half of a circle
centered at the origin
with radius 5. What
length and width
should be the
rectangle so that its
area is a maximum?
Area  2 x 25  x 2
x  [0,5]
x  3.536, Area  25
dimension: 7.071 by 3.536
Practice Problem: Cost

A cylindrical can is to hold 20 m3 The
material for the top and bottom costs
$10/m2 and material for the side costs
$8/m2 Find the radius r and height h of the
most economical can.
Practice Problem: Time

A little duckie is in the ocean swimming at a
location 2.5 miles off the coast. A restaurant is
located along the seashore, 4.5 miles down the
coast. The duckie can swim to a point on the
coast, and then run along the coast to the
restaurant. The duckie can swim 1.2 miles/hour,
and can run 1.6 miles/hour. If the restaurant is
in an urgent need to make a roast duck dish,
what is the least amount of time that takes for
the duckie to get to the restaurant?
Practice Problem: Ladder

Find the length of the
shortest ladder that
will reach over an 8-ft
high fence to a large
wall which is 3 ft
behind the fence.
Practice Problem: Angle

What angle between two edges of length 3 will
result in an isosceles triangle with the largest
area?
Practice Problem: Viewing Angle

A movie screen on a
wall is 20 feet in
height and 10 feet
above the floor. At
what distance x from
the front of the room
should you position
yourself so that the
viewing angle of the
movie screen is as
large as possible?
Practice Problem: Total Area

Four feet of wire is to
be used to form a
square and 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?
Minimize Carbs
Fin…