Question

Communication To set long-distance rates, telephone companies superimpose an imaginary coordinate plane over the United States. Each ordered pair on this coordinate plane represents the location of a telephone exchange. The phone company calculates the distances between the exchanges in miles to establish long-distance rates. Suppose two exchanges are located at \((53,187)\) and \((129,71)\). What is the distance between these exchanges to the nearest mile? The location units are in miles.

Short answer

The distance is approximately 139 miles.

Step-by-step solution

Understand the Problem

We are asked to find the distance between two points, which represent telephone exchanges, on a coordinate plane. The points given are \((53,187)\) and \((129,71)\).

Use the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For our points, \((x_1, y_1) = (53, 187)\) and \((x_2, y_2) = (129, 71)\).

Substitute the Values into the Formula

Substitute the point values into the distance formula: \[ d = \sqrt{(129 - 53)^2 + (71 - 187)^2} \]

Calculate the Differences

Calculate the differences: \(x_2 - x_1 = 129 - 53 = 76\) \(y_2 - y_1 = 71 - 187 = -116\)

Square the Differences

Square the differences calculated:\((76)^2 = 5776\) \((-116)^2 = 13456\)

Sum the Squares

Add the squares of the differences:\(5776 + 13456 = 19232\)

Take the Square Root

Find the square root of the sum to get the distance:\(d = \sqrt{19232} \approx 138.6\)

Round to the Nearest Mile

Round the distance to the nearest mile: 139 miles.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, known as coordinates. Think of it as a map that helps you find or represent locations. It's like looking at a city grid with intersecting lines where each location has a unique address.

On this plane, there are two main lines:

• The horizontal line called the x-axis.
• The vertical line called the y-axis.

These axes divide the plane into four sections or quadrants. By using the coordinate plane, we can easily find the position of any point using its x and y coordinates. In our exercise, the telephone exchanges are placed on this plane so they can be pinpointed accurately.

Ordered Pairs

Ordered pairs are sets of two numbers that represent points on the coordinate plane. They are always written in the form i.e., i.e., i.e.,\((x, y)\), where \(x\) is the number from the x-axis and \(y\) is the number from the y-axis. You can think of them as addresses for each point.

In our example, the ordered pairs are \((53, 187)\) and \((129, 71)\). The first number corresponds to the x-coordinate, and the second number to the y-coordinate. This structure is essential because it provides us a specific position on the plane, allowing for precise navigation and computation, such as calculating the distance between points.

Distance Calculation

Distance calculation on a coordinate plane involves using the distance formula. This formula is a handy tool for finding the length between two points. It helps you calculate real-world distances and is especially useful in math, geography, and various applications.

The distance formula is derived from the Pythagorean theorem and looks like this:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]To use the formula:

• Subtract the x-coordinates to get the difference in the horizontal direction.
• Subtract the y-coordinates to get the difference in the vertical direction.
• Square both differences.
• Add the results.
• Take the square root to find the distance.

In our scenario, this formula lets telephone companies calculate how far apart phone exchanges are, ensuring they set accurate rates based on the real distance. With the given points, the calculation yielded 139 miles, which means that precise rates can now be set for calls between these exchanges.