Math 7030

Activity 10 - Optimization

Optimization refers to the general process of maximizing or minimizing certain quantities. One of the important features of calculus is the ability to solve a large variety of optimization problems. A large number of optimization problems can be handled by a different method that is based on the geometric-arithmetic mean inequality.

Consider a set of positive numbers, say 2,3,6,8,9. The arithmetic mean of these numbers is the ordinary average

A different way of averaging these numbers is by the geometric mean, which is defined as

For a set of n positive numbers, the geometric mean is the n^th root of the product of the numbers. If a number is repeated in the set of numbers, then that number occurs in the product for each occurrence in the list.

The geometric-arithmetic mean inequality states for any set of positive numbers, the geometric mean is always less than or equal to the arithmetic mean and equality occurs if and only if all the numbers in the list are equal.

There is an interesting algorithm for going from the original list of numbers to the geometric mean. We illustrate this with the set of numbers 2,4,7,8,9. The algorithm is to repeat the following step until all the numbers in the list are equal: replace the smallest number x with the arithmetic mean A and replace the largest number y by x + y -A.

For the set of numbers 2,4,7,8 and 9, we have A = 6 and so our new set is 4,5,6,7,8. Note that the arithmetic mean remains fixed at 5, but the product of the five numbers has gone from 2592 to 6720. Repeating the algorithm gives the new set 5,5,6,7,7. The product of these numbers is 7350. Applying the algorithm again gives the set 5,6,6,6,7 with product 7560. Finally, we apply the algorithm one last time producing the set 6,6,6,6,6 with product 7776. Note that each time we applied the algorithm, the product of the set of numbers increased. In particular we have the original product

2*4*7*8*9 < 6*6*6*6*6

If we take fifth roots of both sides, we get

On the left we have the geometric mean of our original set whereas on the right we have the arithmetic mean of the original set. This verifies the geometric-arithmetic mean inequality for our original set of numbers 2,4,7,8,9.

It is not hard to show that this algorithm proves the result. Suppose we have a list of numbers with arithmetic mean A, largest number x and smallest number y. We replace y by A and x by x + y - A. Since A + (x + y - A) = x + y, the sum and hence the arithmetic mean of the new set is the same. On the other hand A(x + y - A) - xy = (A - x)(y - A) ³ 0 since the arithmetic mean of any set of numbers lies between the largest and smallest numbers in the set. This implies A(x + y - A) ³ xy and the inequality is strict unless the largest number in the set equals the smallest number in the set in which case the set consists of a single number. This shows that regardless of what set of numbers we have, whenever we apply the algorithm, the product of the numbers increases and there the geometric mean increases. Eventually every number gets replaced by A and so the geometric mean of the original set is less than or equal to A and equality occurs if and only every number in the original set is the same.

1. Consider the following set S of numbers: 1,3,5,7,9,17. Apply the above algorithm until the set consists of a single number and verify that the geometric means increase each time the algorithm is applied.

2. A rectangle is inscribed within a right triangle as shown below. The length of AB is 3 while the length of BC is 2. We want to determine the values of x and y that give the rectangle of maximum area.

a. Express the area of the rectangle in terms of x and y

b. Express y in terms of x using similar triangles

c. Show that the area is a constant times the product of two numbers having a constant sum.

d. The result on geometric and arithmetic means implies that the maximum area occurs when these numbers are the same.

e. Use this fact to solve for x and y.

f. What is the maximum area?

g. Verify this experimentally with GSP.

3. A trapezoid is inscribed inside a semicircle of radius 1 as shown below.

Use GSP to approximate the maximum area and perimeter of such a trapezoid.

4. A circle of radius 1 and 2 intersect so that the circle of radius 2 passes through the center of the circle of radius 1.

Use GSP to approximate the maximum area and perimeter of a rectangle inscribed between the two circles as shown above. Assume the rectangle has sides parallel and perpendicular to the line joining the two centers of the circles. Does the maximum area and perimeter occur with the same rectangle?

5. With GSP construct a triangle and experimentally try to find the point inside the triangle for which the sum of the distances to the vertices of the triangle is minimal. The point within the triangle that minimizes the sum of the distances to the vertices is called the Fermat point of the triangle. Here is a construction of the Fermat point. Construct an equilateral triangle on each side of the triangle. Then construct line segments between each vertex of the original triangle and farthest vertex of the equilateral triangle constructed on the opposite side.

The line segment is one such line. The three line segments are in fact concurrent at the Fermat point. How close did your experimental point come to the point of intersection of these three lines? Each of the three line segments drawn to determine the Fermat point determines a cevian of the original triangle. Verify Ceva's theorem for these three cevians. (Actually Ceva's theorem can be used to prove that the three line segments are concurrent. This does not prove that the Fermat point minimizes the sum of the distances to the three vertices of the triangle.)

Bonus

The following problem is difficult to solve even with calculus. However you can get a good approximation to this difficult problem with GSP. This will test your skill with GSP.

The question is for a given set of halls that meet at right angles and a box having a given length, what is the maximum width of a box that can be moved around the corner from one hall to the other.

You want to design this sketch with maximum flexibility. In particular, you should be able to change the sizes of the halls. You should also be able to change the length and width of the box. For a given set of hall widths and a given box you should be able to see if the box gets around the corner simply by dragging a point in GSP.