3.4. APPLIED OPTIMIZATION
193
More applied optimization problems
Many of the steps in Preview Activity 3.4 are ones that we will execute in any applied
optimization problem. We briefly summarize those here to provide an overview of our
approach in subsequent questions.
• Draw a picture and introduce variables. It is essential to first understand what
quantities are allowed to vary in the problem and then to represent those values
with variables. Constructing a figure with the variables labeled is almost always
an essential first step. Sometimes drawing several diagrams can be especially
helpful to get a sense of the situation. A nice example of this can be seen at
http://gvsu.edu/s/99, where the choice of where to bend a piece of wire into
the shape of a rectangle determines both the rectangle’s shape and area.
• Identify the quantity to be optimized as well as any key relationships among the
variable quantities. Essentially this step involves writing equations that involve the
variables that have been introduced: one to represent the quantity whose minimum
or maximum is sought, and possibly others that show how multiple variables in the
problem may be interrelated.
• Determine a function of a single variable that models the quantity to be optimized;
this may involve using other relationships among variables to eliminate one or
more variables in the function formula. For example, in Preview Activity 3.4, we
initially found that V = x 2 y, but then the additional relationship that 4x + y = 108
(girth plus length equals 108 inches) allows us to relate x and y and thus observe
equivalently that y = 108 − 4x. Substituting for y in the volume equation yields
V (x) = x 2 (108 − 4x), and thus we have written the volume as a function of the single
variable x.
• Decide the domain on which to consider the function being optimized. Often the
physical constraints of the problem will limit the possible values that the independent
variable can take on. Thinking back to the diagram describing the overall situation
and any relationships among variables in the problem often helps identify the
smallest and largest values of the input variable.
• Use calculus to identify the absolute maximum and/or minimum of the quantity
being optimized. This always involves finding the critical numbers of the function
first. Then, depending on the domain, we either construct a first derivative sign
chart (for an open or unbounded interval) or evaluate the function at the endpoints
and critical numbers (for a closed, bounded interval), using ideas we’ve studied so
far in Chapter 3.
• Finally, we make certain we have answered the question: does the question seek the
absolute maximum of a quantity, or the values of the variables that produce the
193
More applied optimization problems
Many of the steps in Preview Activity 3.4 are ones that we will execute in any applied
optimization problem. We briefly summarize those here to provide an overview of our
approach in subsequent questions.
• Draw a picture and introduce variables. It is essential to first understand what
quantities are allowed to vary in the problem and then to represent those values
with variables. Constructing a figure with the variables labeled is almost always
an essential first step. Sometimes drawing several diagrams can be especially
helpful to get a sense of the situation. A nice example of this can be seen at
http://gvsu.edu/s/99, where the choice of where to bend a piece of wire into
the shape of a rectangle determines both the rectangle’s shape and area.
• Identify the quantity to be optimized as well as any key relationships among the
variable quantities. Essentially this step involves writing equations that involve the
variables that have been introduced: one to represent the quantity whose minimum
or maximum is sought, and possibly others that show how multiple variables in the
problem may be interrelated.
• Determine a function of a single variable that models the quantity to be optimized;
this may involve using other relationships among variables to eliminate one or
more variables in the function formula. For example, in Preview Activity 3.4, we
initially found that V = x 2 y, but then the additional relationship that 4x + y = 108
(girth plus length equals 108 inches) allows us to relate x and y and thus observe
equivalently that y = 108 − 4x. Substituting for y in the volume equation yields
V (x) = x 2 (108 − 4x), and thus we have written the volume as a function of the single
variable x.
• Decide the domain on which to consider the function being optimized. Often the
physical constraints of the problem will limit the possible values that the independent
variable can take on. Thinking back to the diagram describing the overall situation
and any relationships among variables in the problem often helps identify the
smallest and largest values of the input variable.
• Use calculus to identify the absolute maximum and/or minimum of the quantity
being optimized. This always involves finding the critical numbers of the function
first. Then, depending on the domain, we either construct a first derivative sign
chart (for an open or unbounded interval) or evaluate the function at the endpoints
and critical numbers (for a closed, bounded interval), using ideas we’ve studied so
far in Chapter 3.
• Finally, we make certain we have answered the question: does the question seek the
absolute maximum of a quantity, or the values of the variables that produce the