Student Resources
Optimization: Power lines
This activity is meant as a group discussion, a group quiz, or a take home assessment with opportunity for revision based on feedback.
Task One:
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 city wants to lay the new line from the power station, located on 1st Avenue, to the shopping center on Shopping Lane. Due to existing buildings, the power line must be underground in the region east of 1st Avenue, but it can be built above ground along 1st Avenue. The additional cost of burying the power line doubles the cost per mile of installing it. The shopping center is 10 miles south and 7 miles east of the station, as depicted in the applet below.
In this applet, you may alter the slider to change the location of the “Switch Point” (i.e., the location along 1st St. at which you begin burying the power line) along 1st St
Task One: Find the location along 1st St. at which the Switch Point should be placed to minimize the cost of running the power line from the Power Station to the Shopping Center.
Task Two: We will now analyze a more general version of the problem in Task One.
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 city wants to lay the new line from the power station, located on 1st Avenue to the shopping center on Shopping Lane. Due to existing buildings, the power line must be underground in the region east of 1st Avenue, but it can be built above ground along 1st Avenue. The cost per mile of laying out the power line aboveground is \(A\), and the cost per mile of burying the wire is \(U\). The shopping center is \(M\) miles south and \(N\) miles east of the station, as depicted in the applet below.
Note: While all parameters (\(M\), \(N\), \(A\), \(U\)) can affect the optimization process, you should consider them to be fixed when differentiating to determine the cost-minimizing location of the switch point.
Task: Answer the following questions and justify your answers
Questions
In all of the questions below, \(D_0\) refers to the specific distance from the 1st St - Shopping Lane intersection at which placing the switch point minimizes the cost of running the wire.
- What is the relationship (if there is one) between \(N\) and \(D_0\)? For instance, does increasing or decreasing \(N\) always change \(D_0\) in a certain way? Is there a consistent pattern or formula that tells us how \(N\) and \(D_0\) are related? Do \(N\) and \(D_0\) also depend on other parameters of the problem?
- What is the relationship (if there is one) between \(M\) and \(D_0\)? For instance, does increasing or decreasing \(M\) always change \(D_0\) in a certain way? Can you determine a consistent pattern or formula that tells us how \(M\) and \(D_0\) are related? Do \(M\) and \(D_0\) also depend on other parameters of the problem?
- For what values of \(M\), \(N\), \(U\), and \(A\) is it most beneficial to run the power line entirely underground?
- For what values of \(M\), \(N\), \(U\), and \(A\) is it most beneficial to run the power line above ground to the end of 1st St. before burying the power line?