1. What is the relationship (if there is one) between M and D0? For instance, does increasing or decreasing M always change D0 in a certain way? Can you determine a consistent pattern or formula that tells us how M and D0 are related? Do M and D0 also depend on other parameters of the problem?

You are the city planner in charge of running a high- efficiency power line from a power station to a new shopping center being built nearby. The power station is located on 1st St, and the shopping center is on Shopping Ln. The power station is M miles north of Shopping Ln, and the shopping center is N miles east of 1st St. While the power line can run above ground along 1st St, it must be run underground away from 1st St. In addition to the variable location of the switch point, there are three other variable parameters to consider for this scenario: 1) the cost relationship between running a power line above ground versus running the line below ground (i.e., in Prompt 1 it cost three times as much to run the wire underground, but perhaps it costs five times as much, or perhaps it costs D' as much), 2) the distance (M) from the power station to Shopping Ln, and 3) the distance (N) from the shopping center to 1st St. While all four parameters can affect the optimization process, the only variable whose rates of change should be considered for optimization is the variable distance between Shopping Lane and the switch point. As such, once the values for the cost relationship, M, and N are determined, they should be considered constant when differentiating to determine the cost-minimizing distance between the switch point and the power station. In the next GeoGebra application, you may alter the "D=" slider as before to change the location of the viewed switch point. You may also alter the "M=" and "N=" sliders to view how these parameters change the boundaries of the optimization process. The "M=" and "N=" sliders stop at a value of 100, but for the sake of the questions below you may should M and N to be unbounded. I = 5 Switch Point M = 10 Wire above Power Station ground on 1st St. N = 7 10 miles Wire below ground 7 miles Shopping Center on Shopping La. Task: Answer the following questions and justify your answers. CD