So I have this math project that I cannot understand and I would really appreciate it if someone could help me out and explain how I'm supposed to figure it out.
I work for a cable TV company and 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. This home has a 2 mile long driveway 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. It IS possible to 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. (I got this part)
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. (This is where the trouble starts for me!)
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 household \$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?
If you could answer any of these parts that would be amazing! Thank you so much for the help!! 😄
Best Answer
B) For $0\le x\le 5$, the cost incurred from the highway part of the cable is $100\cdot x$ dollars. The length of line on customer premises can be obtained via Pythagoras as $\sqrt{2^2+(5-x)^2}=\sqrt{29-10x+x^2}$, so the corresponding cost is $140\cdot \sqrt{29-10x+x^2}$ dollars and the total cost is $$ 100\cdot x+140\cdot \sqrt{29-10x+x^2}\text{ dollars.} $$
You can plug in $x=0,1,2,3,4,5$ to build the table in part C.