As seismological station is located at (0,-4) 4 kilometers away from a straight shoreline where the x-axis runs through. The epicenter of the earthquake was determine to be 10 kilometers away from the station.
a. Find the equation of the curve that contains the possible location of the epicenter.
b. If furthermore, the epicenter was determined the be 7 kilometers away from the shore, find it's coordinates.
Sorry can u collaborate more of your answer. I don't get itðŸ˜
a. The epicenter is 4 kilometers below the seismological station, so the equation of the curve that contains the possible location of the epicenter would be a parabola with vertex at (0, -4) and opens upwards. The general equation of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Plugging in the coordinates of the vertex, we get y = a(x - 0)^2 - 4. Since the epicenter is 10 kilometers away from the station, we need to find the value of a. Plugging in (10, 0) into the equation, we get 0 = a(10 - 0)^2 - 4. Solving for a, we have a(100) = 4, so a = 4/100 = 1/25. Therefore, the equation of the curve is y = (1/25)x^2 - 4.
b. If the epicenter is 7 kilometers away from the shore, we need to find the coordinates of the point on the curve that is 7 units above the x-axis. Plugging in y = 7 into the equation, we have 7 = (1/25)x^2 - 4. Solving for x, we get (1/25)x^2 = 11, x^2 = 275, and x = sqrt(275) or approximately 16.583. So the coordinates of the epicenter would be (16.583, 7).
To solve the problem, we will use the distance formula to represent the relationship between the epicenter and the station.
a. Equation of the curve:
Let's assume the epicenter's coordinates on the curve are (x, y). Since we know that the seismological station is located at (0, -4) and the epicenter is 10 kilometers away from the station, we can use the distance formula:
√((x - 0)^2 + (y - (-4))^2) = 10
Simplifying this equation gives us:
√(x^2 + (y + 4)^2) = 10
To find the equation of the curve, we square both sides of the equation:
x^2 + (y + 4)^2 = 10^2
Thus, the equation of the curve that contains the possible location of the epicenter is:
x^2 + (y + 4)^2 = 100
b. Coordinates of the epicenter:
If the epicenter is 7 kilometers away from the shore, we can modify the equation above.
Since the x-axis is the shoreline, the y-coordinate of any point on the shoreline is 0. Since the epicenter is 7 kilometers away from the shore, we can replace y with 0 in the equation.
x^2 + (0 + 4)^2 = 100
x^2 + 16 = 100
x^2 = 100 - 16
x^2 = 84
Taking the square root of both sides:
x = ±√84
Therefore, the coordinates of the epicenter are (√84, 0) and (-√84, 0).
Thank you
recall that the equation for a circle with center at (h,k) and radius r is
(x-h)^2 + (y-k)^2 = r^2
Here, you have the station as the center, and the radius of the circle is 10.
(a) So just plug in your numbers.
(b) Now pick a point on that circle with y-coordinate of -7 (why not +7?)