Mathematics 131 -- Calculus for Physical and Life Sciences 1, section 1
Discussion 4 -- Applied Optimization (Max/Min) Problems
November 16, 2007
Background
Yesterday, we discussed some first examples of optimization
or ``max/min'' problems. Many of these come from more or
less realistic applications. A general strategy for solving these:
- Step 1: Draw pictures illustrating several different possible
solutions of the problem, if appropriate.
- Step 2: Identify which quantities are changing in your
different solutions (the variables), and name them. Write down the quantity
to be maximized or minimized, giving it a name too.
- Step 3: (If there is more than one variable), write down any relations
between the variables, and use the relation to solve for all variables
in terms of one of them.
- Step 4: Substitute from results of Step 3 into the function giving the
quantity from Step
2. The goal here is to obtain a function of only one variable.
- Step 5: Find the critical numbers of the function from Step 4.
- Step 6. Classify critical critical as local maxima or local minima using
First or Second Derivative Tests. If the variable is limited
to an interval, determine the function values at the endpoints, and
find the overall maximum or minimum as indicated in the problem.
(Note: ``largest, biggest, greatest, etc.'' in the statement of
the problem usually means you are looking for a maximum value, while
``smallest, least, cheapest, etc.'' usually indicates you are looking for
a minimum.)
- Step 7. Find the maximum or minimum and write down the final answer.
(And, of course, be sure you are answering the question that
was asked!)
Today, we want to practice using this on several examples.
Discussion Questions
- A) A metal can with a top is to be manufactured to contain 30 cubic inches
of Mamma Mia super-spicy pizza sauce. The can will have the shape of a
right circular cylinder. Find the dimensions (height and radius of the
cylinder) that will minimize the
cost of the can, assuming the metal costs .04 cents per square inch.
- B) For some species of birds, it takes more energy to
fly over water than over land (over land,
they can make use of updrafts). A lesser tufted grebe (this is a species of bird)
leaves an island 5 km from point A, the nearest point
to the island on a long straight shore. The grebe's nest is at
point B, 13 km along the shore from point A.
If it takes 1.4 times as much energy to fly one km over water
as it does to fly one km over land, where on the shoreline should
the bird head first in order to minimize the total energy needed
for the flight from the island to the nest.
- C) Northern Iowa State Agricultural and Veterinary Junior College
is building a new running track for their prize-winning track team --
the ``Flying Farmers''. The track is to be the perimeter of a region obtained
by putting two semicircles on opposite ends of a rectangle, and that
perimeter should be 440 yards in length. Due to
budget problems, the administration has decided to grow
sweet corn in the area enclosed by the track and sell it to a local
grocery store to generate some extra revenue. Determine the dimensions
to build the track in order to maximize the area for growing corn.
Assignment
Group write-ups due in class Tuesday, November 20.