Investigation The retail outlets described in Exercise 20 are located at and (see

Chapter 13, Problem 21

(choose chapter or problem)

Investigation The retail outlets described in Exercise 20 are located at (0, 0), (4, 2), and (-2, 2) (see figure). The location of the distribution center is (x, y), and therefore the sum of the distances S is a function of x and y.

(a) Write the expression giving the sum of the distances S. Use a computer algebra system to graph S. Does the surface have a minimum?

(b) Use a computer algebra system to obtain \(S_{x}\) and \(S_{y}\). Observe that solving the system \(S_{x}=0\) and \(S_{y}=0\) is very difficult. So, approximate the location of the distribution center.

(c) An initial estimate of the critical point is \(\left(x_1,\ y_1\right)=(1,\ 1)\). Calculate \(-\nabla S(1,\ 1)\) with components \(-S_{x}(1,\ 1)\) and \(-S_{y}(1,\ 1)\). What direction is given by the vector \(-\nabla S(1,\ 1)\)?

(d) The second estimate of the critical point is

\(\left(x_2,\ y_2\right)=\left(x_1-S_x\left(x_1,\ y_1\right)t,\ y_1-S_y\left(x_1,\ y_1\right)t\right)\).

If these coordinates are substituted into S(x, y), then S becomes a function of the single variable t. Find the value of t that minimizes S. Use this value of t to estimate \(\left(x_2,\ y_2\right)\).

(e) Complete two more iterations of the process in part (d) to obtain \(\left(x_4,\ y_4\right)\). For this location of the distribution center, what is the sum of the distances to the retail outlets?

(f) Explain why \(-\nabla S(x,\ y)\) was used to approximate the minimum value of S. In what types of problems would you use \(\nabla S(x,\ y)\)?

Text Transcription:

S_x

S_y

S_x = 0

S_y = 0

(x_1, y_1) = (1, 1)

- nabla S(1, 1)

- S_x (1, 1)

- S_y (1, 1)

(x_2, y_2) = x_1 - S_x (x_1, y_1)t, y_1 - S_y (x_1, y_1)t)

(x_2, y_2)

(x_4, y_4)

- nabla S(x, y)

nabla S(x, y)