Answer Happy

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 difference in cost 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), 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 variables can affect the optimization process, the only variable whose rates of change should be considered for optimization is the variable distance between the power station and the switch point. As such, once the values for the cost difference, 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.
5:38 Discussion Details MATH 241 Calculus & Anyl Geometry I-May 2022-Onl... minimizing distance between the switch point and the power station. In the next GeoGebra application, you may alter the "x="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. M = 10 N=7 2=5 Switch Point Shopping Center on Shopping Ln. ◄ Previous 888 Dashboard Calendar Wire above Power Station ground on 1st St. Wire below ground 7 miles To Do 10 miles Notifications Next > Inbox
1. 1) What is the relationship (if there is one) between the optimal switch point - Power Station distance and the distance from the Shopping Center to 1st St. - Shopping Ln. intersection? i.e., how does changing N change the cost-minimizing distance? 2. What is the relationship between the optimal switch point - Power Station distance and the distance from the Power Station to 1st St. - Shopping Ln. intersection? i.e., how does changing M change the cost-minimizing distance? o Note: Example relationships might include operations on N and the switch point distance, such as "The optimal switch point distance is x=N-2" or "The optimal switch point distance is proportional to N", or there may be no relationship at all. General statements such as "Decreasing N also decreases the switch point distance" or "Increasing N decreases the switch point distance" may also apply. 3. Under what conditions should the power line be immediately run underground? i.e., determine a range of values for M and N, and determine the cost relationships that would make running the wire entirely underground the cost-minimizing option. o Note: Example relationships might include operations on M and the switch point, such as "The optimal switch point is located at x=M-2" or "The optimal switch point is proportional to M", or there may be no relationship at all. General statements such as "Decreasing N also decreases the switch point location" or "Increasing N decreases the switch point location" may also apply. 4. Under what conditions should the power line be run above ground until reaching Shopping Ln? i.e., determine a range of values for M and N, and determine the cost relationships that would make running the wire entirely above ground along 1st St. the cost-minimizing option.