Optimization Problems
Optimization Problems
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An optimization problem seeks to find the largest
(the smallest) value of a quantity (such as maximum
revenue or minimum surface area) given certain
limits to a problem.
An optimization problem can usually be expressed as
“find the maximum (or minimum) value of some
quantity Q under a certain set of given conditions”.
Steps in Solving Optimization Problems:
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Understand the Problem
Draw a Diagram
Introduce Notation identifying the quantity to be
maximized (or minimized), Q, and the other variables
(label the diagram)
Express Q as a function of the other variables. Also
express any relations or conditions among the other
variables with equations.
Rewrite Q as a function of one variable, using given
relationships and determine the domain D of Q.
Find the absolute extrema (maximum of minimum) of
Q on domain D (use the Closed Interval Method, if the
domain of Q is an closed interval)
The Least Expensive Cable
Better Cable Company must provide service to a customer
whose house is located 2 miles from the main highway. The
nearest connection box for the cable is located 5 miles down
the highway from the customers’ driveway.
The installation cost is $14 per mile for any cable that is laid
from the house to the highway. (The cable may be laid along
the driveway to the house or across the field).
The cost is $10 per mile when the cable is laid along the
highway.
Determine where the cable should be laid so that the
installation cost as low as possible.
Look at the picture:
2 miles
Connection Box
5 miles
How much will the customer have to pay if the cable is
laid 5 miles along the highway and 2 miles along the
drive to the house?
Show your calculations.
Do you think this cable will be the least expensive
possibility?
Explain your reasoning.
Calculus in England
No summer visit to England is complete without having lunch on
the sunny Goodge Street. There, for a mere pound coin, you can
purchase the best fish and chips you’ve ever tasted from any one
of friendly street vendors.
One of the reason that the prices are so reasonable is that they
give you no silverware, nor even a plate: they just roll up a piece
of paper into a cone, and toss your food in. (They do give you a
little packet of vinegar, though).
Neither a long, skinny cone nor a wide fat cone would hold enough
fish and chips to make anybody happy. The vendors must be trained
to roll a cone of a perfect size.
Many students will never get a chance to see sunny Goodge Street,
but by trying to solve the vendors’ problem of optimizing the volume
of a cone, we can feel as if we are there now.
For modeling purposes, assume that the piece of paper is a
circle of radius 5 inches, and that we are cutting a wedge out
of it whose central angle is Θ.
Θ
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5
Find the maximum volume of this cone.
The Waste-Free Box
There is a traditional problem that goes like this:
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We want to make an open-topped box from an 8,5 x 11 inch
sheet of paper by cutting congruent squares from the corners
and folding up the sides. What is the maximum possible
volume of such a box?
What most people never think about is the fate of those four
squares of paper. They don’t have to be wasted. By taping them
together, and putting the reluctant structure on a desk, one can
make a handsome pen-and-pencil holder, which will be a box with
neither top nor bottom. (It will still hold pencils as long as it rests
on the desk).
Picture:
x
x
x
x
x
x
8,5
x
bottom
x
11
no bottom
1. What is the maximum possible combined volume of an
open-topped box plus a handsome pen-and pencil holder
that can be made by cutting four squares from an 8,5 x 11
inch sheet of paper?
2. Describe the open-topped box that results from the
maximal case. Intuitively, why do we get the result we do?
3. Repeat this problem for a 6 x 10 inch piece of paper.