Mathematics 136 -- AP Calculus, section 4
Discussion 2 -- Applied Optimization (Max/Min) Problems
October 6, 2003
Background
Last week, you 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 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. 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, 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 examples.
Discussion Questions
- A) 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.
- B) 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
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 generate some extra revenue. Determine the dimensions
to build the track in order to maximize the area for growing corn.
- C) The cost of fuel to propel a boat through the water (in dollars
per hour) is proportional to the cube of the speed. The Loon
Island ferry boat Crazy Asa
uses $100 of fuel per hour when cruising at 10 miles per hour.
Apart from fuel there are also $675 per hour costs for the crew's
wages, docking fees, upkeep on the boat, etc. What should
the cruising speed be in order to minimize the total cost per mile traveled.
(Hint: How can you get dollars per mile from dollars per hour and
miles per hour?)
- D) You have a 10 inch length of flexible metal wire. If you
cut the wire into two pieces, bend one to form a square, and bend
the other to form a circle, what is the largest total area
(square plus circle) that you can enclose with your two bent wires?
Give a full justification of your answer.
Assignment
Group write-ups due in class on Wednesday, October 8.