You work for Silver Satellite & Cable TV Company in Research & Development Department. You have been inquired to come up with formula to find cost of running cable from connection box to new cable household. First example that you are working with engages Steven family, who own rural home with driveway 2 miles long extending to the house from nearby highway. Nearest connection box is along highway but 5 miles from driveway.
It costs company $100 per mile to install cable along highway and $140 per mile to install cable off highway. As Steven's house is surrounded by farmland which they own, it would be possible to run cable overland to house directly from connection box or from any point between connection box to driveway.
i) Draw the sketch of problem situation, suppose that highway is straight road and driveway is also straight road perpendicular to highway. Incorporate two or more possible routes for cable.
ii) Let x represent distance in miles that cable runs along highway from connection box before turning off toward house. State total cost of installation as function of x. (you may like select toanswer part (c) before part (b) if you would like to analyze concrete instances before creating equation.)
iii) Create the table of possible integral values of x and corresponding cost in every instance. Does one choice appear to cost least?
iv) If you charge Stevens $800 for installation, woud you be eager to let them select which way cable would go? Describe.
v) using the graphing calculator, graph function from part (b) and find value of x which would make installation cost minimum.
vi) Before proceeding further with installation, you check local regulations for cable companies and determine that there is pending state legislation which says that cable can't turn off highway more than 0.5 mile from Steven's driveway. If the legislation passes, find ultimate cost of installing Steven's cable?
vii) If cable company wants to install cable in 5000 homes in the area, and suppose that figures for Steven's installation are typical, how much will new legislation cost company overall if they can't use cheapest installation cost, but instead have to follow new state regulations?