Hyperbola: define epicenter word problem

[ducmod]

Homework Statement

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!

Homework Equations

The Attempt at a Solution

 

[Samy_A]

Homework Statement

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!

Homework Equations

The Attempt at a Solution

The primary wave travels from the hypocenter to A, from the hypocenter to B, from the hypocenter to C, not from A to B.

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?

These two statements are not the same.

 

[ducmod]

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.