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Hyperbola: define epicenter word problem

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  1. Mar 4, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello!
    Here is the word problem that should be solved based on hyperbola equation (exercise from
    hyporbola topic):
    The P-waves (\P" stands for Primary) of an earthquake
    in Sasquatchia travel at 6 kilometers per second.10 Station A records the waves rst. Then
    Station B, which is 100 kilometers due north of Station A, records the waves 2 seconds later.
    Station C, which is 150 kilometers due west of Station A records the waves 3 seconds after
    that (a total of 5 seconds after Station A). Where is the epicenter?

    My question is about the first part, i.e. station A and B (for now):

    Given the distance between station A and station B of 100 km,
    and the speed of waves of 6 km/second, and the fact that
    station A has recorded the wave 2 seconds earlier than station B,
    how can the distance between them be 100 km?

    Please, help me to understand this and correct me:
    wave speed is 6 km/sec, hence in 2 seconds it covered 12 kilometers.
    Shouldn't it mean that the distance between A and B is 12 km, which is
    the same as to say that the epicenter is 12 kilometers closer to A than
    to B?


    Thank you!
    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Mar 4, 2016 #2

    Samy_A

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    Science Advisor
    Homework Helper

    The primary wave travels from the hypocenter to A, from the hypocenter to B, from the hypocenter to C, not from A to B.
    These two statements are not the same.
     
    Last edited: Mar 4, 2016
  4. Mar 4, 2016 #3
    I think I got it. Please, take a look at my results and let me know if they are correct, and if not, please, guide me to find my mistakes:

    1) station B is located due north from station A and both are at the foci; distance between them is 100 km.
    Assume the center of hyperbola is at (0, 0), hence the equation is y^2 / b^2 - x^2 / a^2 = 1.
    c = 50, coordinate of station A is (0, -50), B (0, 50)
    b = 6 (half of the difference in the distance that the sound had to travel to B, i.e. 2 seconds * 6 km/sec = 12, half of it is 6)
    (it is a vertical hyperbola, hence it's be, not a, that is equal to 6).
    a^2 = c^2 - b^2 = 2464
    thus equation for this *vertical* hyperbola between A and B stations is y^2 / 36 - x^2 / 2624 = 1

    2) station C is 150 due west from station A, hence it lies at (-150, -50),
    The hyperbola between station C and A is a horizontal one, and given the difference in wave detection
    of 3 seconds, a = 9.
    c = 75. center at (-75, -50)
    b^2 = c^2 - a^2 = 5544
    equation (x + 75)^2 / 81 - (y + 50)^2 / 5544 = 1

    The epicenter lies closer to A in both cases, hence the point will be in the 4th quadrant, closer to A,
    at the intersection of lower part of A-B hyperbola and right part of A-C hyperbola.
     
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