Cost of Cable

Question

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 the 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 integer 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 the derivative of the function, 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 miles 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?

Answer 1

(a) I placed the connection box at (0, 0), the entrance to the driveway at (5, 0), and the house at (5, 2). The highway is the horizontal line between the connection box and the driveway. The driveway is perpendicular to the highway. Possible routes include running the cable directly from the box to the house, leaving the highway at a point between the box and the driveway, or following the highway to the driveway and then going to the house.

(b) If x is the distance traveled along the highway, the remaining horizontal distance to the driveway is 5 - x. The off-highway cable forms the hypotenuse of a right triangle with legs 5 - x and 2.

Off-highway distance = sqrt((5 - x)^2 + 4)

The total cost is therefore:

C(x) = 100x + 140sqrt((5 - x)^2 + 4), where 0 <= x <= 5.

(c) The costs for the integer values of x are:

x = 0: $753.92

x = 1: $726.10

x = 2: $704.78

x = 3: $695.98

x = 4: $713.05

x = 5: $780.00

Of the integer choices, x = 3 miles gives the lowest cost.

(d) Yes. Every route represented by the function costs less than $800. The most expensive choice costs $780, so the company would still make at least $20.

(e) The derivative of the cost function is:

C'(x) = 100 - 140(5 - x)/sqrt((5 - x)^2 + 4)

Set the derivative equal to zero:

100 = 140(5 - x)/sqrt((5 - x)^2 + 4)

Solving gives:

x = 5 - 5/sqrt(6), or approximately 2.96 miles.

The cable should travel about 2.96 miles along the highway before turning toward the house. The minimum installation cost is approximately $695.96.

(f) Under the proposed law, the cable must leave the highway within 0.5 mile of the driveway. Therefore, x must be at least 4.5. Since the cost is increasing over the permitted interval from x = 4.5 to x = 5, the least expensive permitted choice is x = 4.5.

C(4.5) = 100(4.5) + 140sqrt((0.5)^2 + 2^2)

C(4.5) is approximately $738.62.

(g) The additional cost per home is approximately:

$738.62 - $695.96 = $42.66

Using the unrounded amounts for 5,000 homes gives:

5,000($738.6174 - $695.9592) = $213,291.07

The legislation would therefore cost the company approximately $213,291 more for 5,000 installations.