Mathematics 133 -- Intensive Calculus for Science 1

Discussion 6 -- Applied Optimization (Max/Min) Problems

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. 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 additional examples.

A) 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 make some extra income. Determine the dimensions to build the track in order to maximize the area for growing corn.
B) 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!)
C) 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.
D) A cylindrical can with a top is made to contain 30 cubic inches of condensed tomato soup. 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.

Group write-ups due in class on Wednesday, November 28. I'll return these the next day -- Thursday, November, 29.