Hyperbola: define epicenter word problem

• ducmod
In summary, the problem involves determining the location of the epicenter of an earthquake in Sasquatchia based on the time difference of wave detection at three different stations. By solving for the hyperbolas between the stations, it is determined that the epicenter is closer to station A, at the intersection of the lower part of the A-B hyperbola and the right part of the A-C hyperbola.
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!

ducmod said:

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!

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.
ducmod said:
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.

Last edited:
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.

1. What is a hyperbola?

A hyperbola is a type of geometric curve that is defined by the equation (x-h)^2/a^2 - (y-k)^2/b^2 = 1. It is a two-dimensional shape that resembles two curved lines that open up and out from each other.

2. What is the epicenter of a hyperbola?

The epicenter of a hyperbola is the point at which the two curved lines of the hyperbola intersect. It is located at the center of the hyperbola and is the point of symmetry for the shape.

3. How is the epicenter of a hyperbola related to its foci?

The epicenter of a hyperbola is equidistant from both of its foci. This means that the distance from the epicenter to each focus is the same, and this distance is known as the semi-major axis of the hyperbola.

4. How can hyperbolas be used to solve real-world problems?

Hyperbolas can be used to solve real-world problems in a variety of fields, including physics, engineering, and economics. For example, hyperbolas can be used to model the trajectory of a particle in motion or the relationship between supply and demand in market economics.

5. How can the epicenter of a hyperbola be determined in a word problem?

To determine the epicenter of a hyperbola in a word problem, you will need to first identify the coordinates of the foci and a point on the hyperbola. Then, you can use the distance formula to find the distance between the foci and the point, and use this information to calculate the coordinates of the epicenter.

• Precalculus Mathematics Homework Help
Replies
1
Views
6K
• Precalculus Mathematics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
2
Views
3K
• Introductory Physics Homework Help
Replies
3
Views
2K
• Introductory Physics Homework Help
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
23
Views
490
• Mechanics
Replies
27
Views
3K
• Special and General Relativity
Replies
31
Views
3K
• Precalculus Mathematics Homework Help
Replies
2
Views
2K
• Precalculus Mathematics Homework Help
Replies
3
Views
2K