Optimization
Steps for Solving Optimization Problems
1. Draw the picture and label variables.
2. Determine a constraint equation (if necessary) and a maximizing
(minimizing) equation.
3. Use the constraint equation to solve for one of the variables and substitute it
into the maximizing (minimizing) equation.
4. Take a derivative and set it equal to zero. Then solve.
5. Answer the question.
Examples
A. 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 = r2h
The cost equation gives us our optimization equation
Cost = 2r2(.1) + 2rh(.09)
The volume equation gives us:
h = 150/r2
so that
Cost of top and bottom + Cost of sides
C = .2r2 + .18r(150/r2)
= .2r2 + 27/r
To find the minimum cost we take the derivative and set it equal to 0:
C' = 4r - 27/r2 = 0
So that
4r3 = 27
or
r3 = 27/(4

r = 2.14 in
so that
h = 150/(2.142) =10.4 in
B. 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
TimeBeach = x/15
and
TimeWater =
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. 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?
B. 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.