On-line Math 21

4.3 Max-min problems

These problems are a more serious introduction to mathematical modeling than the related rates problems. Many legitimate questions in science and engineering center on optimization, finding the best possible configuration of a given situation. In engineering, many of these questions relate to the costs involved (directly or indirectly), and in science many models of natural systems have as their base assumptions that certain energies are minimized.

As with related rates problems, the point for us will be to reduce the question to the mathematical essence, usually expressed as a function f(x) measuring the quantity we want to optimize. Then, usually, you find the optimum value of x by differentiating f and solving f¢(x) = 0 for x . You will have to keep in mind that maxima/minima (critical points) occur either at a place where f¢(x) = 0 , or an endpoint of the interval of definition, or at a place where the derivative doesn't exist. You also have to keep in mind that certain quantities (length, width, area, volume) have to he greater than or equal to 0. This often determines the interval of definition of the function.

Finally, many situations will start off the description with a quantity which you have to optimize, which has more than one independent variable. There will then be ``constraints'', equations among the variables, that you use to solve one variable in terms of another, to get to a function of one variable.

Example 1 A cardboard box

Find the dimensions of the largest cardboard box with a square base which uses only 12 square feet of cardboard. For a cardboard box, remember that the top and bottom have two layers of cardboard (for the flaps), while the sides are a single layer.

Remark This is typical of real problems, in that the problem is poorly phrased. Largest how? Largest in volume.

Solution

Example 2 The fence

Farmer Egbert wants to fence off some of his farmyard for a pen for his chickens. He has 100 feet of fencing, and plans to place one side of the pen against the side of his barn. How should he design his pen? Like most pens, this one is supposed to be rectangular. He wants to give his chickens as much room as possible.

Solution

Exercise 1 The Norman window

Architect U. R. Pay is designing a new house. It features a ``Norman window'' in the foyer. A Norman window is shaped like a regular window, as a rectangle, except that the top is semi-circular. Because the frame is the primary expense, the contractor bills the customer based on the length of the perimeter, at $30 per foot. On the other hand, the light that the window lets into the room is proportional to the area of glass. Find the proportions for the window to get the most light for a fixed amount of money.

Answer:

Exercise 2 Another box

Take a 12 inch by 15 inch rectangle of tin, cut equal squares out of each corner and bend up the sides, then solder the seams on the edges to make a rectangular box with no top. How large a square should you cut out of each corner to maximize the volume of the resulting box?

Answer:

Example 3 Little Red Riding Hood

Little Red Riding Hood is on her way to Grandma's house. Red lives 10 miles down the (straight) road from Grandma's mailbox, which is the closest point on the road to Grandma's house. The house itself is Ö3 miles from the road, in the woods. Now, Red can walk 4 miles per hour on the road, but only 2 miles per hour in the woods, since she has to keep looking out for the Big Bad Wolf. What Red plans to do is, at some point between her house and Grandma's mailbox, to leave the road and walk in a straight line to Grandma's house, through the woods. Where should she leave the road, if she wants to get to Grandma's house in the least amount of time?

Solution

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On 8 Dec 2000, 00:23.