Cost of Cable You work for the Silver Satellite & Cable TV Company in the Resear
ID: 3032578 • Letter: C
Question
Cost of Cable You work for the Silver Satellite & Cable TV Company in the Research & Development Department. You have been asked to come up with a formula to determine the cost of running cable from a connection box to a new cable household. The first example that you are working with involves the Steven family, who own a rural home with a driveway 2 miles long extending to the house from a nearby highway. The nearest connection box is along the highway but 5 miles from the driveway. It costs the company $100 per mile to install cable along the highway and $140 per mile to install cable off the highway. Because the Steven’s house is surrounded by farmland that they own, it would be possible to run the cable overland to the house directly from the connection box or from any point between the connection box to the driveway.
(a) Draw a sketch of this problem situation, assuming that the highway is a straight road and the driveway is also a straight road perpendicular to the highway. Include two or more possible routes for the cable.
(b) Let x represent the distance in miles that the cable runs along the highway from the connection box before turning off toward the house. Express the total cost of installation as a function of x. (You may choose to answer part (c) before part (b) if you would like to examine concrete instances before creating the equation.)
(c) Make a table of the possible integral values of x and the corresponding cost in each instance. Does one choice appear to cost the least?
(d) If you charge the Stevens $800 for installation, would you be willing to let them choose which way the cable would go? Explain.
(e) Using a graphing calculator, graph the function from part (b) and determine the value of x that would make the installation cost minimum. (f) Before proceeding further with the installation, you check the local regulations for cable companies and find that there is pending state legislation that says that the cable cannot turn off the highway more than 0.5 mile from the Steven’s driveway. If this legislation passes, what will be the ultimate cost of installing the Steven’s cable?
(g) If the cable company wishes to install cable in 5000 homes in this area, and assuming that the figures for the Steven’s installation are typical, how much will the new legislation cost the company overall if they cannot use the cheapest installation cost, but instead have to follow the new state regulations?
I have all the other answers, but i need a very detailed graph for e.
I also need a very detailed explanation for f.
please help
Explanation / Answer
a) You're on your own for a drawing but that shouldn't be too hard.
b) If x is the length along the highway, then the "hypotenuse" from the road to the house is sqrt([5 - x]^2 + 2^2) = sqrt(x^2 - 10x + 29). The total cost function is 100x + 140sqrt(x^2 - 10x + 29). To minimize this, find the derivative and where it equals 0 (0<=x<=5) is the cheapest point. This would be 100 + 140(1/sqrt(x^2 - 10x + 29))(2x- 10); now, where does this equal 0? Using a spreadsheet, I found at x = 4.235 mi
d) 140sqrt(29) = 753.92 and 100(5) + 140(2) = 780, so whatever route taken to the house would cost less than the quote; please the customer.
e) Did that above.
f) Cost with the regulation would be 450 + 140sqrt(4.5) = 746.98.
g) Cost under the cheapest solution is 100(4.235) + 140sqrt(4.055225) = 705.43; the legislation would cost about $207,800.