MATH 131-503 Fall 2015
4.6
c
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Liu
4.6 Optimization Problems
Steps in Solving Optimization Problems
1. Understand the problem: What is the 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 these symbols.
4. Express Q in terms of some of the other symbols from Step 3
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 f .
6. Use the methods of Sections 4.2 and 4.3 to find the absolute maximum or minimum value of
f . In particular, if the domain of f is a closed interval, then the Closed Interval Method in
Section 4.2 can be used.
Examples:
1. (p. 300) A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders
a straight river. He needs no fence along the river. What are the dimensions of the field that
has the largest area?
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MATH 131-503 Fall 2015
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Liu
2. (p. 301) A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will
minimize the cost of the metal to manufacture the can.
3. (p. 303) Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.
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MATH 131-503 Fall 2015
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Wen
Liu
4. A box with a square base and open top must have a volume of 62500 cm3 . Find the dimensions
of the box that minimize the amount of material used.
5. The top and bottom margins of a poster are each 3 cm and the side margins are each 2 cm. If
the area of printed material on the poster is fixed at 96 cm2 , find the dimensions of the poster
with the smallest area.
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MATH 131-503 Fall 2015
4.6
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Wen
Liu
6. Find the dimensions of a rectangle with area 512000 m2 whose perimeter is as small as possible.
7. Find two positive numbers satisfying the given requirements: the sum of the first and twice the
second is 80 and the product is a maximum.
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