Rough Algorithm for Applied Extremum Problems

  1. Assign letters to the varying quantities, especially to the quantity to be "extremized" (say y).
  2. Express y in terms of other quantities.
  3. Relate the other quantities with equations until they are all functions of one of them (say x).
  4. Substitute the last step into the second-last step to get y = f(x).
  5. Identify the endpoints of the possible values of x.
  6. Find dy/dx and the x-values at which it is 0 or undefined.
  7. Decide which of the candidates in the last two steps give(s) the desired extremum.

Examples of Applied Extremum Problems

1. A variable line through the point (1,2) intersects the x-axis at A(a,0) and the y-axis at B(0,b), where both a and b are positive. Find the area of the triangle AOB of least area. (O is the origin.) Answer

2. A right triangle of given hypotenuse h is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume. Answer

3. An 8-foot fence is 1 foot from a vertical wall. What is the shortest ladder that can pass over the fence and reach the wall? And where should the base of the ladder be?

4. A wire of length L is cut into two pieces; one is bent into a square and the other into an equilateral triangle. How should the wire be cut if the sum of the areas is to be (a) a minimum? (b) a maximum? Answer

5. Suppose a couple can walk 4 meters per second and row 2 m/sec, and a picnic ground is 70 m down and across a canal 30 m wide. To what point on the opposite bank should they row to get to the picnic ground as quickly as possible?

6. The intensity of illumination at any point is proportional to the product of the strength of the light source and the inverse (reciprocal) of the square of the distance from the source. If two sources of light, of relative strengths a and b are at distance c apart, at what point on the line joining them will the intensity be a minimum? Assume the intensity at any point is the sum of the intensities from the two sources. Partial answer

7. A chemist knows that two quantities (say x and y) vary inversely, i.e., y=k/x, but the constant k is not known. What value of k will minimize the sum of squares of the vertical distances from the graph of y=k/x to the chemist's experimental data points (1,5), (2,4) and (4,1)? Answer

Answers

1. 4.

2. 2h3/9(3)1/2

4. (a) 4L/(4+3(3)1/2) for the square.
4. (b) All for the square.

6. x/(c-x) = (a/b)1/3, where x is the distance from the source of strength a

7. k=116/21, about 5.52