3.4. APPLIED OPTIMIZATION
191
critical number
3.4 Applied Optimization
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• In a setting where a situation is described for which optimal parameters are sought,
how do we develop a function that models the situation and use calculus to find
the desired maximum or minimum?
Introduction
Near the conclusion of Section 3.3, we considered two examples of optimization problems
where determining the function to be optimized was part of a broader question. In
Example 3.4, we sought to use a single piece of wire to build two geometric figures (an
equilateral triangle and square) and to understand how various choices for how to cut the
wire led to different values of the area enclosed. One of our conclusions was that in order
to maximize the total combined area enclosed by the triangle and square, all of the wire
must be used to make a square. In the subsequent Activity 3.9, we investigated how the
volume of a box constructed from a piece of cardboard by removing squares from each
corner and folding up the sides depends on the size of the squares removed.
Both of these problems exemplify situations where there is not a function explicitly
provided to optimize. Rather, we first worked to understand the given information in the
problem, drawing a figure and introducing variables, and then sought to develop a formula
for a function that models the quantity (area or volume, in the two examples, respectively)
to be optimized. Once the function was established, we then considered what domain was
appropriate on which to pursue the desired absolute minimum or maximum (or both). At
this point in the problem, we are finally ready to apply the ideas of calculus to determine
and justify the absolute minimum or maximum. Thus, what is primarily different about
problems of this type is that the problem-solver must do considerable work to introduce
variables and develop the correct function and domain to represent the described situation.
Throughout what follows in the current section, the primary emphasis is on the reader
solving problems. Initially, some substantial guidance is provided, with the problems
progressing to require greater independence as we move along.
Preview Activity 3.4. According to U.S. postal regulations, the girth plus the length of a
parcel sent by mail may not exceed 108 inches, where by “girth” we mean the perimeter of
the smallest end. What is the largest possible volume of a rectangular parcel with a square
191
critical number
3.4 Applied Optimization
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• In a setting where a situation is described for which optimal parameters are sought,
how do we develop a function that models the situation and use calculus to find
the desired maximum or minimum?
Introduction
Near the conclusion of Section 3.3, we considered two examples of optimization problems
where determining the function to be optimized was part of a broader question. In
Example 3.4, we sought to use a single piece of wire to build two geometric figures (an
equilateral triangle and square) and to understand how various choices for how to cut the
wire led to different values of the area enclosed. One of our conclusions was that in order
to maximize the total combined area enclosed by the triangle and square, all of the wire
must be used to make a square. In the subsequent Activity 3.9, we investigated how the
volume of a box constructed from a piece of cardboard by removing squares from each
corner and folding up the sides depends on the size of the squares removed.
Both of these problems exemplify situations where there is not a function explicitly
provided to optimize. Rather, we first worked to understand the given information in the
problem, drawing a figure and introducing variables, and then sought to develop a formula
for a function that models the quantity (area or volume, in the two examples, respectively)
to be optimized. Once the function was established, we then considered what domain was
appropriate on which to pursue the desired absolute minimum or maximum (or both). At
this point in the problem, we are finally ready to apply the ideas of calculus to determine
and justify the absolute minimum or maximum. Thus, what is primarily different about
problems of this type is that the problem-solver must do considerable work to introduce
variables and develop the correct function and domain to represent the described situation.
Throughout what follows in the current section, the primary emphasis is on the reader
solving problems. Initially, some substantial guidance is provided, with the problems
progressing to require greater independence as we move along.
Preview Activity 3.4. According to U.S. postal regulations, the girth plus the length of a
parcel sent by mail may not exceed 108 inches, where by “girth” we mean the perimeter of
the smallest end. What is the largest possible volume of a rectangular parcel with a square