Problem 38 - Algebra 2

Question

The Rose Bowl is located about 35 miles west and about 40 miles north of downtown Los Angeles. Suppose an earthquake occurs with its epicenter about 55 miles from the stadium. Assume that the origin of a coordinate plane is located at the center of downtown Los Angeles. Write an equation for the set of points that could be the epicenter of the earthquake.

Step-by-Step Solution

Answer

The equation is \((x + 35)^2 + (y - 40)^2 = 3025\).

Step 1: Understand the Problem

We have two known locations: the Rose Bowl and the epicenter of the earthquake. The Rose Bowl is located 35 miles west (negative x-direction) and 40 miles north (positive y-direction) from downtown Los Angeles, which we consider as the origin. The epicenter is 55 miles from the Rose Bowl.

Step 2: Represent the Rose Bowl as Coordinates

Since the Rose Bowl is 35 miles west and 40 miles north, its coordinates are (-35, 40) on the Cartesian plane.

Step 3: Formulate the Equation of the Circle

The epicenter forms a circle centered at the Rose Bowl with a radius of 55 miles. The general equation for a circle is \((x - h)^2 + (y - k)^2 = r^2\), where (h, k) is the center and r is the radius. Here, the center is the Rose Bowl at (-35, 40), and the radius is 55.

Step 4: Write the Equation

Plugging in the values into the circle's equation yields: \((x + 35)^2 + (y - 40)^2 = 55^2\).Simplify the radius: \(55^2 = 3025\).

Step 5: Final Equation of the Circle

Thus, the final equation representing the set of points that could be the epicenter of the earthquake is: \((x + 35)^2 + (y - 40)^2 = 3025\).

Key Concepts

Cartesian Plane

The Cartesian Plane is a fundamental concept in coordinate geometry. It consists of two perpendicular axes, typically labeled as the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.

The point where the axes intersect is termed the origin and is denoted by (0, 0). Each point on the plane is represented by a pair of numbers, known as coordinates, which specifies its distance from the origin along the x and y axes.

In our problem, the downtown Los Angeles is taken as the origin point. This allows us to easily locate the Rose Bowl garage, placed at (-35, 40). The negative sign in the coordinate indicates the direction is west (left) of the origin, while the positive number signifies north (up). Understanding this system is crucial for solving problems in coordinate geometry.

Equation of Circle

In coordinate geometry, the equation of a circle becomes a powerful tool to define points at a uniform distance from a central point. For a circle centered at point \(h, k\) with a radius \(r\), the equation used is \( (x - h)^2 + (y - k)^2 = r^2 \).

This equation captures the essence of a circle—a set of all points equidistant from a given point (the center).

In our context, the Rose Bowl acts as the center, and knowing this simplifies finding the earthquake's potential epicenter as it lies on a circle with radius of 55 miles.

Substituting the Rose Bowl's coordinates (-35, 40) and the radius 55 into the equation forms the specific circle \( (x + 35)^2 + (y - 40)^2 = 3025 \), illustrating every possible location of the epicenter evenly distanced at 55 miles from the stadium.

Distance Formula

The distance formula is another crucial concept in coordinate geometry, enabling us to find the straight-line distance between two points on a Cartesian Plane. It is expressed as: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \], where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.

In our exercise, while the primary focus was on forming the equation of a circle, underlying calculations rely on understanding distance as well. Here, it confirmed the relationship between the Rose Bowl and potential epicenter points—all 55 miles away, ensuring the calculated circle's radius accurately reflects the problem's parameters.

This formula reinforces precision in solving spatial problems by providing a method to calculate or verify distances in various scenarios.