Two separate seismograph stations receive indication of an earthquake in the form of a wave traveling to them in a straight line from the epicenter and shaking the ground at their locations. Station B is 60 due east of station A. The epicenter is located due north of station A and 45 north of due west from station B.
Part A
Determine the distance from the epicenter to A.
Express your answer using two significant figures.
Part B
Determine the distance from the epicenter to B.
Express your answer using two significant figures.
Part C
Station C is located an additional 10 east of B. At what angle does C report the direction of the epicenter to be?
To solve this problem, we can use the principles of trigonometry and geometry. Let's go step by step to find the answers to each part of the question:
Part A: Determining the distance from the epicenter to A.
Since the epicenter is located directly north of station A, and we know the distance between A and B, we can use the Pythagorean theorem to find the distance from the epicenter to A. Let's call this distance "x."
We can create a right triangle with sides x, 60, and the hypotenuse being the direct line distance between A and B. Since we have a right triangle, we can use the Pythagorean theorem which states:
(AB)^2 = (AC)^2 + (BC)^2
Solving for x:
x^2 = (60)^2 + (AB)^2
x^2 = 3600 + (AB)^2
To find AB, we can use the distance formula between two points:
AB = √[(change in x)^2 + (change in y)^2]
In this case, the change in x is 60 and the change in y is 45 (since the epicenter is 45 degrees north of due west from station B). Substituting these values back into the equation:
x^2 = 3600 + (√[(60)^2 + (45)^2])^2
x^2 = 3600 + (3600 + 2025)
x^2 = 3600 + 5625
x^2 = 9225
Taking the square root of both sides, we get:
x = √9225
x ≈ 96
Therefore, the distance from the epicenter to A is approximately 96 units (the actual units are not specified in the question, so we can't determine the exact distance).
Part B: Determining the distance from the epicenter to B.
Since station B is directly east of the epicenter, the distance from the epicenter to B is the same as the distance from A to B, which is 60 units (as given in the question).
Therefore, the distance from the epicenter to B is 60 units.
Part C: Determining the angle at which C reports the direction of the epicenter.
To find the angle at C, we can use basic trigonometry principles. Station C is located an additional 10 units east of B. If we draw a line from B to the epicenter, and another line from C to the epicenter, we can see that we have a right triangle where the adjacent side is 10 units (the additional distance between B and C), and the opposite side is the distance from the epicenter to B (which we found to be 60 units in Part B).
Using the tangent function, we can find the angle:
tan(θ) = opposite/adjacent
tan(θ) = 60/10
θ = tan^(-1)(60/10)
θ ≈ 81.87
Therefore, station C reports the direction of the epicenter to be at an angle of approximately 81.87 degrees.