Mathematics 133 -- Intensive Calculus for Science 1
Discussion 7 -- Applied Optimization (Max/Min) Problems
November 15, 2005
Background
Yesterday we discussed some first examples of applied optimization
or ``max/min'' problems. We had 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 one variable
in terms of the other.
- 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. 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.
It is often helpful also to classify the critical points
as local maxima or local minima using First or Second Derivative Tests.
(Note: ``largest, biggest, greatest, etc.'' in the statement of
the problem usually means you are looking for a maximum value, while
``smallest, least, etc.'' usually indicates you are looking for
a minimum.)
- Step 7. Find maximum or minimum and write down the final answer.
Today, we want to practice using this on several additional examples.
Discussion Questions
- A) Two non-negative numbers x,y add up to 72.
What is the largest possible value of the product P = xy?
(In the outline above, think of starting with Step 3, since the
first two steps are either irrelevant or done already by the statement
of the problem!)
- B) A rectangular beam is cut from a cylindrical log of radius
30 cm. The strength of a beam of width w and height h
is proportional to wh2. Find the width
and the height of the beam of maximum strength.
- C) Northern South Dakota
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
financial constraints, the administration has decided to grow
sweet corn in the area enclosed by the track and sell it to the local
grocery store to make some extra income. Determine the dimensions
to build the track in order to maximize the area for growing corn.
Assignment
Group write-ups due in class on Wednesday, November 16. I'll return
these the next day -- Thursday, November, 17.