D)x^2+y^2=625 X^2+(y-50)^2=1122 Y=20 (-15,20)(15,20)

E)x^2+y^2=625 X^2+y^2-200y+10000=13618 Y=18 (-20,-15)(20,-15)

1)Combine your answers from Parts D and E to pinpoint the epicenter of the earthquake

2)Graph the equations for all three stations to verify your algebraic solution.

you can graph the three circles at wolframalpha . com by typing

plot x^2+y^2=625, x^2+(y-50)^2=1122, x^2+y^2-200y+10000=13618

but they don't intersect in a common point. Not sure how to interpret the results.

To pinpoint the epicenter of the earthquake, we can use a method called trilateration. Trilateration requires at least three coordinates of stations (in this case, coordinates from parts D and E) and the distances from each station to the epicenter.

To find the distances, we can use the equations of the circles centered at each station. The formula for the distance between two points (x1, y1) and (x2, y2) is:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

1) Let's find the distances between each station and the epicenter by substituting the given coordinates into the equations:

Station D: (x, y) = (-15, 20)
Distance from Station D: sqrt((-15)^2 + (20 - 20)^2) = 15

Station E: (x, y) = (-20, -15)
Distance from Station E: sqrt((-20)^2 + (-15 - 18)^2) ≈ 43.2

2) Now we have the distances from two of the stations. To find the third distance, we can use the coordinates from the equation in part D:

Station D (part D equation): (x, y) = (15, 20)

Distance from Station D (part D equation): sqrt((15)^2 + (20 - 20)^2) = 15

Based on our calculations, the distances are:

Station D: 15
Station E: 43.2
Third Station (based on part D equation): 15

Since we have all three distances, we can now pinpoint the epicenter using the trilateration method.

To verify the algebraic solution, we can graph the equations for all three stations:

Graph of Station D (x^2 + y^2 = 625):
This equation represents a circle centered at the coordinates (0, 0) with a radius of 25 units.

Graph of Station E (x^2 + y^2 - 200y + 10000 = 13618):
This equation represents a circle centered at the coordinates (0, 100) with a radius of approximately 100 units.

Now compare the graphs of the circles for Station D, Station E, and the equation from part D. The epicenter of the earthquake will be the point where all three circles intersect.