Question

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)

Short answer

Answer: In more complex scenarios involving multiple houses with different coordinates, the optimal location of a power substation found by minimizing the sum of actual distances would likely be different from the one found by minimizing the sum of squared distances. This is because the two minimization methods consider different factors when determining the optimal location.

Step-by-step solution

Calculate the mean of x and y coordinates separately

In this case, we have the following coordinates: -house 1: (0, 0) -house 2: (2, 0) -house 3: (1, 1) Calculate the mean of x coordinates and mean of y coordinates: mean_x = (0 + 2 + 1) / 3 = 1 mean_y = (0 + 0 + 1) / 3 = 1/3 So, the optimal location of the power substation is (1, 1/3). #Part b - General case of n=3: Houses at (x1, y1), (x2, y2), and (x3, y3)#

Calculate the mean of x and y coordinates separately

In this case, we have the following coordinates: -house 1: (x_1, y_1) -house 2: (x_2, y_2) -house 3: (x_3, y_3) Calculate the mean of x coordinates and mean of y coordinates: mean_x = (x_1 + x_2 + x_3) / 3 mean_y = (y_1 + y_2 + y_3) / 3 So, the optimal location of the power substation is (mean_x, mean_y). #Part c - General case of n houses: at (x1,y1), (x2,y2),..., (xn,yn)#

Calculate the mean of x and y coordinates separately

In the general case, we have the following coordinates: -house 1: (x_1, y_1) -house 2: (x_2, y_2) ... -house n: (x_n, y_n) Calculate the mean of x coordinates and mean of y coordinates: mean_x = (x_1 + x_2 + ... + x_n) / n mean_y = (y_1 + y_2 + ... + y_n) / n So, the optimal location of the power substation is (mean_x, mean_y). #Part d - Comparison of sum of squares and sum of distances#

Write the function giving sum of distances

We'll use the locations in part (a): (0,0), (2,0), and (1,1) Let the power substation be located at point P (x, y). The sum of distances from P to the houses will be the following function: distance_sum = sqrt((x-0)^2 + (y-0)^2) + sqrt((x-2)^2 + (y-0)^2) + sqrt((x-1)^2 + (y-1)^2)

Use a graphing utility to determine the optimal location

Now, use a graphing utility (e.g., Desmos, Geogebra) to plot the function distance_sum and determine whether the optimal location is the same as in part (a). The comparison using the graphing utility will show that the optimal location of the substation by minimizing the sum of squared distances is not always the same as the one obtained by minimizing the sum of actual distances. In the case of part (a), the optimal location remains the same. However, in more complex scenarios, the locations will likely be different.

Key concepts

Mean calculation

One of the fundamental concepts in optimization problems is calculating the mean of coordinates. This is commonly utilized to find an optimal central location. In the exercise, when we have coordinates of houses, we find the mean of the x-coordinates and y-coordinates separately. For example, for the points

• (0, 0)
• (2, 0)
• (1, 1)

the mean of the x-coordinates is \[\text{mean}_x = \frac{0 + 2 + 1}{3} = 1\]
and for y-coordinates,\[\text{mean}_y = \frac{0 + 0 + 1}{3} = \frac{1}{3}. \]
Thus, the optimal point is \((1, \frac{1}{3})\). Calculating the mean helps find a point that minimizes the sum of squared distances to each house, providing an efficient and straightforward solution.

Coordinate geometry

Coordinate geometry plays a crucial role in understanding the layout of various points on a plane. In this exercise, each house is considered a point with specific \((x, y)\) coordinates. Understanding how these coordinates work helps visualize the problem and formulate solutions. The optimal location for the substation involves finding a point that minimizes overall distances among given coordinates. By using concepts from coordinate geometry like calculating distances and means, you can efficiently analyze and solve these types of problems.
This understanding allows us to see the relationship between different points on the coordinate plane. It forms the basis for calculating distances and eventually minimizing those distances, leading to the best possible placement of the substation.

Distance minimization

Minimizing distances is a critical step in various optimization problems. In the given exercise, the goal is to place a power substation such that the sum of squared distances to the houses is minimized. This involves calculating the Euclidean distance between each house and the proposed substation location. The formula for distance from a house at \((x_i, y_i)\) to a point \((x, y)\)is \(\sqrt{(x - x_i)^2 + (y - y_i)^2}.\)
However, since the exercise involves minimizing the sum of squares of these distances, the formula adjusts to simpler calculations, as squares eliminate negative values and focus purely on magnitude, simplifying mathematical manipulation. This approach typically leads to simpler calculations compared to minimizing the sum of actual distances.

Steiner's problem

Steiner's problem, also related to distance minimization, involves finding points that minimize the total distance to a given set of points. However, unlike the exercise problem, Steiner's problem seeks to minimize the sum of direct distances rather than squared distances.For part d of the exercise, the provided locations at

• (0, 0)
• (2, 0)
• (1, 1)

are used to construct a function representing the sum of distances \(d(x, y) = \sqrt{(x - 0)^2 + (y - 0)^2} + \sqrt{(x - 2)^2 + (y - 0)^2} + \sqrt{(x - 1)^2 + (y - 1)^2}.\) Using graphing utilities to plot this function can help visualize and find optimal solutions. While the solution for minimizing squared distances may coincide with direct distances in simple scenarios, more complex arrangements might reveal different optimal locations, reflecting the nuanced nature of Steiner’s problem.