Problem 36 - Algebra 2

Question

Edison is located at (9, 3) in the coordinate system on a road map. Kettering is located at (12, 5) on the same map. Each side of a square on the map represents 10 miles. To the nearest mile, what is the distance between Edison and Kettering?

Step-by-Step Solution

Answer

The distance between Edison and Kettering is 36 miles.

Step 1: Identify Coordinates

First, identify the coordinates of Edison and Kettering from the problem statement. Edison is at \((9, 3)\) and Kettering is at \((12, 5)\).

Step 2: Apply the Distance Formula

Use the distance formula to find the distance between the two points. The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Substitute the Coordinates

Substitute the coordinates into the distance formula: \(d = \sqrt{(12 - 9)^2 + (5 - 3)^2}\).

Step 4: Simplify Within the Square Root

Calculate the differences and square them: \((12 - 9)^2 = 3^2 = 9\) and \((5 - 3)^2 = 2^2 = 4\). So, the expression becomes \(d = \sqrt{9 + 4}\).

Step 5: Solve the Square Root

Simplify under the square root: \(d = \sqrt{13}\).

Step 6: Convert the Distance

Since each side of a square represents 10 miles, calculate the actual distance: \(\sqrt{13} \times 10 \approx 36.06\).

Step 7: Round to the Nearest Mile

Round 36.06 to the nearest mile to get the final distance: 36 miles.

Key Concepts

Coordinate System

In a coordinate system, each point is identified by a pair of numbers known as coordinates. These coordinates are usually presented as \(x, y\), where \(x\) represents the horizontal position and \(y\) represents the vertical position on a grid. This grid is often two-dimensional, like the one used in the given exercise.

Consider the example from the exercise: Edison is located at \(9, 3\) and Kettering at \(12, 5\). Here, \(9\) and \(12\) are the x-coordinates, while \(3\) and \(5\) are the y-coordinates. Such a system allows easy navigation and calculation of distances between points on a map.

Using coordinate systems:

• Helps to establish positions precisely.
• Facilitates the use of mathematical formulas to calculate distances.
• Enables visualization of points on a grid for better comprehension.

Square Root Calculation

Calculating square roots is a fundamental mathematical operation and an essential step in finding the distance between two points. The square root of a number is a value that, when multiplied by itself, gives the original number. In the distance formula, square roots help to calculate the straight-line distance in a coordinate system.

For instance, in this exercise, we start with the expression \( \sqrt{13} \). Here the square root of 13 is key to determining the distance between Edison and Kettering.

Steps involved in square root calculation:

• Identify the value under the square root (e.g., 13).
• Find its square root value using a calculator or estimation.
• Approximate if necessary for practical purposes, like converting units or rounding.

In practice, learning how to calculate square roots by hand can also enhance your understanding of numbers and improve your numeric intuition. However, when precision is a must, use a calculator.

Distance Between Points

Finding the distance between two points in a coordinate system requires understanding and correctly applying the distance formula. This mathematical tool allows you to determine the shortest path between two given points.

The distance formula is expressed as: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). It involves:

• Subtracting the x-coordinates \(x_2 - x_1\) and y-coordinates \(y_2 - y_1\).
• Squaring both differences to ensure positive values.
• Summing these squares and taking the square root of the result.

This formula gives the direct "as-the-crow-flies" distance, effectively using the Pythagorean theorem.

In our exercise, substituting the coordinates of Edison \(9, 3\) and Kettering \(12, 5\) into the distance formula results in \( d = \sqrt{(12 - 9)^2 + (5 - 3)^2} = \sqrt{13} \). Which, when adjusted for scale as in our context, converts to a more practical transportation distance, rounded to 36 miles.