Optimization
Steps for Solving Optimization Problems

Draw the picture and label variables.

Determine a constraint equation (if necessary) and a maximizing
(minimizing) equation.

Use the constraint equation to solve for one of the variables and
substitute it into the maximizing (minimizing) equation.

Take a derivative and set it equal to zero. Then
solve.

Answer the question.
Examples

You want to construct a can that holds 150 cubic inches of
juice as cheaply as possible. The top and bottom costs .1 cents per square
inch and the side costs .09 cents per square inch. What should the dimensions
of the can be?
Solution:
First we use the volume of a cylinder to get the constraint equation
150 = pr^{2}h
The cost equation gives us our optimization equation
Cost = 2pr^{2}(.1)
+ 2prh(.09)
Cost of top and bottom + Cost of sides
The volume equation gives us:
h = 150/pr^{2}
so that
C = .2pr^{2} +
.18pr(150/pr^{2})
= .2pr^{2 }+ 27/r
To find the minimum cost we take the derivative and set it equal to 0:
C' = 4pr  27/r^{2 }= 0
So that
4pr^{3} = 27
or
r^{3} = 27/(4p)
r = 2.14 in
so that
h = 150/p(2.14^{2}) =10.4 in

A lifeguard swims at a rate of 5 feet per second and can run at a rate of
15 feet per second. Suppose that the lifeguard spots a drowning child in
the ocean 200 feet down the shore and 50 feet out at sea. How far should
the lifeguard run until (s)he begins swimming?
Solution
Our goal is to minimize the total transit time. The total transit time
is
Total Transit
Time (T) = Time Along the Beach + Time in the Water
Using
Time =
Distance/Rate
We have
Time_{Beach}
= x/15
and
Time_{Water}
=
Hence
Taking a derivative and setting it equal to 0 gives
After a lot of algebra or using a computer we get
x = 200  25/ @ 182.3 feet
We can conclude that the runner should run a little
more than 182 feet before diving into the water.
Exercises
Hold your mouse over the yellow rectangle for the answer.

A poster is to have an area of 120 square inches with one inch margins
at the bottom and sides and a 2 inch margin at the top. What dimensions
will give the largest printed area?

A quarter mile race track is to be designed by having a rectangle
with semicircles on each end. Find the dimensions that will make the
area of the rectangle as large as possible.
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