Problem 73 - Precalculus Essentials

Question

A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake's 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.

Step-by-Step Solution

Answer

Hence, the standard form of the equation representing the possible epicenters for the quake is \((x+2.4)^2 + (y+2.7)^2 = 900\)

Step 1: Identify the center of the circle

From the problem, we are given that the USC is located 2.4 miles west and 2.7 miles south of center in the coordinate system. Since west and south correspond to negative directions in a typical cartesian coordinate system, USC's coordinates will be (-2.4, -2.7). This represents the center of the circle where the seismograph is.

Step 2: Identify the radius of the circle

The exercise states that the earthquake's epicenter was approximately 30 miles away from the University, so the radius of the circle would be 30 miles.

Step 3: Construct the equation of the circle

The standard form of the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where (h,k) are the coordinates of the center of the circle and r is the radius. Using the center coordinates (-2.4, -2.7) and the radius 30, we get \((x+2.4)^2 + (y+2.7)^2 = 900\), where 900 is the square of the radius 30. This is the equation of the circle that represents all possible points for the earthquake's epicenter.

Key Concepts

Understanding the Coordinate System

In mathematics, a coordinate system is a framework used to define and describe positions in a space, allowing us to pinpoint precise locations. One of the most common types of coordinate systems is the cartesian coordinate system. This system uses two or more axes to locate points. For a 2D space like a map or graph, we use the x-axis (horizontal) and y-axis (vertical). These axes intersect perpendicularly at the origin point (0,0). Every point in this space has coordinates

• The x-coordinate tells us the position left or right of the origin.
• The y-coordinate tells us the position above or below the origin.

For example, the University of Southern California's location is given as (-2.4, -2.7) in relation to central Los Angeles, indicating it is 2.4 miles west (negative x-direction) and 2.7 miles south (negative y-direction) of the origin.

Navigating the Cartesian Plane

The cartesian plane is a two-dimensional surface created by the x and y axes, used to graphically represent points, lines, and shapes. It is named after René Descartes, who developed its concept.
On this plane, coordinates are used as described in the coordinate system section. Any point, like (-2.4, -2.7) for USC's location, denotes a specific spot on the cartesian plane.

• The plane is divided into four quadrants.
• Quadrant I: both coordinates positive.
• Quadrant II: x negative, y positive.
• Quadrant III: both coordinates negative, as with USC.
• Quadrant IV: x positive, y negative.

Understanding this layout helps us position and move through points effectively, enhancing our problem-solving skills in geometry and real-world navigation.

Calculating the Distance Between Points

To find the distance between two points on a cartesian plane, we use the distance formula, which is derived from the Pythagorean theorem. This formula is particularly useful in determining how far apart two locations are in a straightforward manner. The formula is \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]where

• (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
• d is the distance between them.

In the scenario given, the epicenter of the earthquake is 30 miles away from the University. To represent this, the equation of a circle with USC as the center and 30 miles as the radius helps visualize all possible locations of the epicenter.