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4.5 Optimization Problems
Steps in solving Optimization Problems
1. Understand the Problem
Ask yourself: What is unknown? What are the given
quantities? What are the given conditions?
2. Draw a Diagram
3. Introduce Notation
- Assign a symbol to the quantity that is to be maximized or
minimized (let’s call it Q for now)
- Also select symbols (a,b,c,…,x,y) for other unknown
quantities and label the diagram with the symbols
- It may help to use initials as suggestive symbols – for
example, A for area, h for height, t for time
4. Express Q in terms of the other symbols from Step 3
Steps in solving Optimization Problems (cont.)
5. If Q has been expressed as a function of more than
one variable in Step 4
- Use the given information to find relationships (in the form
of equations) among these variables
- Then use these equations to eliminate all but one of the
variables in the expression for Q
- Thus Q will be expressed as a function of one variable x,
say Q = f (x). Write the domain of this function
6. Use the methods of Sections 4.1 and 4.3 to find the
absolute minimum or maximum value of f.
In particular, if the domain of f is a closed interval,
then the Closed Interval Method can be used.
A Classic Optimization 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 − 2 x
A = x (^) ( 40 − 2 x )
2
A = 40 x − 2 x
A ′ = 40 − 4 x
0 = 40 − 4 x
4 x = 40
x = 10
A = 10 40 ( − 2 10⋅ )
A =10 20 ( )
2
A = 200 ft
l = 40 − 2 x
w = x w =10 ft
l = 20 ft
Another Optimization Problem
What dimensions for a one liter cylindrical can will
use the least amount of material?
We can minimize the material by minimizing the area.
2
A = 2 π r + 2 π rh
area of ends
lateral area
We need another equation that relates r and h :
2
V = π r h
3 1 L =1000 cm
2 1000 = π r h
2
h
π r
2 2
A r 2 r
π r
A 2 r r
2
A 4 r r
