Earthquake An earthquake was felt up to 81 miles from its...

Earthquake An earthquake was felt up to 81 miles from its epicenter. You were located 60 miles west and 45 miles south of the epicenter. (a) Let the epicenter be at the point $(0,0) .$ Find the standard equation that describes the outer boundary of the earthquake. (b) Would you have felt the earthquake? (c) Verify your answer to part (b) by graphing the equation of the outer boundary of the earthquake and plotting your location. How far were you from the outer boundary of the earthquake?

Earthquake An earthquake was felt up to 81 miles from its epicenter. You were located 60 miles west and 45 miles south of the epicenter.
(a) Let the epicenter be at the point $(0,0) .$ Find the standard equation that describes the outer boundary of the earthquake.
(b) Would you have felt the earthquake?
(c) Verify your answer to part (b) by graphing the equation of the outer boundary of the earthquake and plotting your location. How far were you from the outer boundary of the earthquake?

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Key Concepts

Distance Formula

The distance formula is a mathematical tool derived from the Pythagorean theorem that calculates the distance between any two points in the Cartesian plane. It is given by d = sqrt((x? - x?)² + (y? - y?)²) and is fundamental in problems where the relative positions and distances between points need to be determined.

Equation of a Circle

In analytic geometry, the equation of a circle is used to represent all points that are at a fixed distance, called the radius, from a specific point, known as the center. The standard form of the circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves representing geometric figures and their properties using a coordinate system. This approach enables the translation of geometric problems into algebraic equations, allowing the use of algebraic methods to solve geometric problems.

Graphical Verification

Graphical verification is the process of confirming algebraic or numerical results by visual representation. By graphing equations and points on the coordinate plane, one can visually inspect the accuracy of solutions, such as determining whether a point lies within or outside a particular boundary.

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