Question

Tamika and Matthew are going to hike from Cedar Creek Cave to the Ford Nature Center. Cedar Creek Cave is located 3 kilometers west of the ranger's station. The Ford Nature Center is located 2 kilometers east and 4 kilometers north of the ranger's station. a. Draw a diagram on a coordinate grid to represent this situation. b. What is the distance between Cedar Creek Cave and Ford Nature Center?

Short answer

The distance is \(\sqrt{41}\) kilometers.

Step-by-step solution

Representing Locations on a Coordinate Grid

First, we define the ranger's station as the origin point on the coordinate grid, which is at (0,0). Cedar Creek Cave is 3 kilometers west of the ranger's station, so it is at (-3,0) on the grid. The Ford Nature Center is located 2 kilometers east and 4 kilometers north, which places it at (2,4) on the coordinate grid.

Use the Distance Formula

Now that we have the coordinates of both locations, Cedar Creek Cave at (-3,0) and Ford Nature Center at (2,4), we use the distance formula to find the distance between these two points. The distance formula is given by: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\] where \((x_1, y_1)\) are the coordinates of the first point and \((x_2, y_2)\) are the coordinates of the second point.

Substitute Values into the Distance Formula

Substitute the coordinates of Cedar Creek Cave (-3,0) and Ford Nature Center (2,4) into the distance formula: \[d = \sqrt{(2 - (-3))^2 + (4 - 0)^2}\]. This becomes \[d = \sqrt{(2 + 3)^2 + (4)^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16}\].

Calculate the Distance

Simplify the expression under the square root: \(\sqrt{25 + 16} = \sqrt{41}\). The distance between Cedar Creek Cave and the Ford Nature Center is \(\sqrt{41}\) kilometers.

Key concepts

Distance Formula

The Distance Formula is a powerful tool in coordinate geometry for finding the distance between two points on a coordinate plane. It is derived from the Pythagorean theorem, which relates the square of the length of the hypotenuse of a right-angled triangle to the sum of the squares of the lengths of the other two sides. The formula itself is expressed as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Where:

• \((x_1, y_1)\) are the coordinates of the first point,
• \((x_2, y_2)\) are the coordinates of the second point,
• \(d\) is the distance between these two points.

To use this formula, simply plug in the coordinates of the two points you're interested in. The calculation then provides a numerical value representing the direct distance between them. It’s particularly useful in various applications in geometry, physics, and even for practical day-to-day problem-solving scenarios.

Coordinate Grid

A coordinate grid, often referred to as a Cartesian plane, is a two-dimensional plane formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis). Both axes are usually placed at right angles to each other. The point where they intersect is called the origin, represented as (0,0). This plane is a useful way to visually map locations using coordinates. Each point on this plane is defined by a pair of numerical coordinates:

• The x-coordinate, which shows the horizontal distance from the origin.
• The y-coordinate, indicating the vertical distance from the origin.

In problem-solving, like finding locations such as Cedar Creek Cave and the Ford Nature Center, a coordinate grid helps by allowing for a visual representation of these locations. It shows the relationship and distance between the two points, making spatial problems more concrete and manageable.

Problem Solving

Problem solving in coordinate geometry involves breaking down a problem into manageable steps and using mathematical principles to find a solution. For the given exercise, the steps included: 1. Identifying and defining the problem: Determining the task of finding the distance between two locations given their coordinates. 2. Representing data: Placing the known locations on a coordinate grid using coordinates based off described positions. 3. Applying relevant concepts: Using the Distance Formula to calculate the distance. When engaging in problem-solving within the realm of geometry, it’s crucial to:

• Clearly understand the problem and what is being asked.
• Label and organize your data effectively.
• Select the right mathematical tool or formula for calculation.

This approach makes complex problems more intuitive and easier to solve.

Geometry

Geometry in a broad sense is the branch of mathematics dealing with shapes, sizes, relative positions of figures, and properties of space. In coordinate geometry, also known as analytic geometry, algebraic principles are combined with geometric concepts to solve problems about size and position. By using coordinates and formulas such as the distance formula, the properties of points and lines can be analyzed and understood in greater depth. Geometry facilitates the breakdown of spatial problems based on location and distances, as seen in this exercise. Learning these principles provides a foundation for higher mathematics and can also be applied in real-world tasks, including navigation, engineering, and design. Understanding the basics of geometry helps in visualizing and solving spatial problems more effectively.