Question

a rectangular coordinate system with coordinates in miles is placed with the origin at the center of a city. the figure to the right indicates that a university is located 2.5 miles west and 2.1 miles south of the center of the city. a seismograph on the campus shows that a small earthquake occurred. the quakes epicenter is estimated to be approximately 30 miles from the university. write the standard form of the equation for the set of points that could be the epicenter of the quake. what is the standard form of the equation of the circle? (simplify your answer.)

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step2: Identify the center and radius

The center of the circle (the university's location) is \((h,k)=(- 2.5,-2.1)\) and the radius \(r = 30\) miles.

Step3: Substitute values into the standard form

Substitute \(h=-2.5\), \(k = - 2.1\), and \(r = 30\) into the equation \((x - h)^2+(y - k)^2=r^2\).
We get \((x-(-2.5))^2+(y - (-2.1))^2=30^2\).
Simplify it to \((x + 2.5)^2+(y + 2.1)^2 = 900\).

Answer:

\((x + 2.5)^2+(y + 2.1)^2=900\)