Mathematics 131 -- Calculus for Physical and Life Sciences 1, section 1
Discussion 4 -- Applied Optimization (Max/Min) Problems
November 17, 2004
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 a picture, if appropriate. Name the variables.
- Step 2: Write down the function 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 from Step
2 to obtain a function of one variable.
- Step 5: Find the critical points of the function from Step 4.
- Step 6. Classify critical points 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) 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 (a 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.
- B) A cylindrical can with a top is made to contain 30 cubic inches
of Mamma Mia super-spicy pizza sauce. 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.
- 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 the 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 Monday, November 22.