Mathematics 136 -- Advanced Placement Calculus, section 2

Applied Optimization (Max/Min) Problems

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.

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 (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.
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 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.