Math

Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns.
Centerville is located at (10,0) in the xy-plane, Springfield is at (0,4), and Shelbyville is at (0,-4). To save on the cost of cable, Greedy Cablevision wants to arrange the cable in a Y-shaped configuation, running cable from Centerville to some point (x,0) on the x-axis where it then splits into two branches, one going to Springfield and one to Shelbyville.

Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your conclusion by answering the following questions.

a. To solve this problem we need to minimize the following function of x:

b. We find that f(x) has a critical point at:

c. To verify that f(x) has a minimum at this critical point we evaluate the second derivative f''(x) at this point.f''(critical point) is:

d. Thus the minimum length of cable needed is:

  1. 👍 0
  2. 👎 0
  3. 👁 5,686
  1. As usual, draw a diagram, and it is clear that

    f(x) = (10-x) + 2√(x^2+16)
    f'(x) = -1 + 2x/√(x^2+16)
    f"(x) = 32/(x^2+16)^3/2

    Now the answers are easy to provide

    1. 👍 2
    2. 👎 5
  2. Thank you! I was able to find everything other than the minimum length of the cable. How would you find that?

    1. 👍 0
    2. 👎 0
  3. Am I correct on these?

    1.Does every line have both an x-intercept and a y-intercept?
    Yes.

    2.Does every line have an infinite number of lines that are parallel to the given line?
    Yes.

    Does every line have an infinite number of lines that are perpendicular to the given line?
    No.

    1. 👍 0
    2. 👎 0
  4. the minimum cable length occurs where f'(x) = 0

    So, you need

    2x/√(x^2+16) - 1 = 0
    2x - √(x^2+16) = 0
    2x = √(x^2+16)
    4x^2 = x^2+16
    3x^2 = 16
    x = 4/√3

    So, the minimum cable length is

    f(4/√3) = 10+4√3

    1. 👍 0
    2. 👎 2
  5. vertical lines have no y-intercept
    horizontal lines have no x-intercept
    (unless these lines are in fact x=0 or y=0)

    #2 ok

    #3 also yes. Pick any point on the line (and there are an infinite number of points) and you can draw a perpendicular line through it.

    1. 👍 0
    2. 👎 0
  6. this problem makes me want to di

    1. 👍 2
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Physics

    In a repair shop a truck engine that has mass 429 kg is held in place by four light cables (Figure 1) . Cable A is horizontal, cables B and D are vertical, and cable C makes an angle of 37.1∘ with a vertical wall. Tension in

  2. physics

    A 20kg traffic light hangs midway on a cable between two poles 40meters apart. If the sag in the cable is .4 meters, what is the tension in each side of the cable?

  3. English

    Juanita sent the two __________ reports to __________ yesterday afternoon. A. technicians'; headquarters B. technicians; headquarter's C. technician's; headquarters' D. technicians'; headquarters' A

  4. Algebra 2

    your local cable company charges $29.99/month for basic cable. Premium channels are available for a surcharge of $5.95/channel. You have $70/month budgeted for cable, how many premium channels can you purchase?

  1. geography

    Employment opportunities for more than five hundred people will be lost when New Development Solutions, Inc., closes its headquarters in the city of Nowhere, USA. What are effects that the closing of New Development is likely to

  2. Pre calculus

    The cable suspension bridge hangs in the shape of a parabola. The towers supporting the cable area is 600 lft apart and 200ft high. If the cable, at its lowest, is 40ft above the bridge at its midpoint, how high is the cable 50ft

  3. applications of proportion math

    Centerville's population decreased from 37,065 to 26,102. How do I find the percent of change?

  4. Statistics

    Listed below are amounts​ (in millions of​ dollars) collected from parking meters by a security service company and other companies during similar time periods. Do the limited data listed here show evidence of stealing by the

  1. math

    The New network cell phone company charges $20 a month for service and $0.02 per minute of talking time. The mobile company charges $15 a month for service and $0.03 per minute of talking time.. (q) write expressions for the total

  2. math

    A company has two office buildings that hold 30,000 employees. Their headquarters contains 14 times as many employees as their overseas branch. Therefore, their headquarters contains __?__ employees.

  3. Math

    A cable TV company charges $24.95 per month for basic service and $6.95 per month for each premium channel. Ryans monthly bill is $45.80. How many premium channels does he receive What would be the algebra expression?

  4. Physics

    A 165.8-N bird feeder is supported by three cables as shown in the figure below. Find the tension in each cable. left cable - ??? right cable - ??? bottom cable - ??? www.webassign.net/sercp9/4-p-019.gif

You can view more similar questions or ask a new question.