Problem 62 - College Algebra

Question

A small craft in ake Ontario sends out a distress signal. The coordinates of the boat in trouble were \((49,64) .\) One rescue boat is at the coordinates (60,82) and a second Coast Guard craft \(\mathrm{s}\) at coordinates (58,47) . Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

Step-by-Step Solution

Answer

Rescue Boat 2 will reach the distressed boat the fastest.

Step 1: Understand the Problem

We have two rescue boats located at different coordinates. We need to determine which rescue boat will reach a distressed boat located at coordinates (49, 64) the fastest. To do this, we will calculate the distance from each rescue boat to the distressed boat and compare them.

Step 2: Find Distance Formula

The formula to calculate the distance between two coordinates (x1, y1) and (x2, y2) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). We will apply this formula to find out the distance from each rescue boat to the distressed boat.

Step 3: Calculate Distance for Rescue Boat 1

Rescue Boat 1 is at coordinate (60, 82), and the distressed boat is at (49, 64). Substitute into the distance formula: \[ d_1 = \sqrt{(60 - 49)^2 + (82 - 64)^2} \] Calculate: \[ d_1 = \sqrt{11^2 + 18^2} = \sqrt{121 + 324} = \sqrt{445} \approx 21.1 \] So, the distance for Rescue Boat 1 is approximately 21.1 units.

Step 4: Calculate Distance for Rescue Boat 2

Rescue Boat 2 is at coordinate (58, 47), and the distressed boat is at (49, 64). Substitute into the distance formula: \[ d_2 = \sqrt{(58 - 49)^2 + (47 - 64)^2} \] Calculate: \[ d_2 = \sqrt{9^2 + (-17)^2} = \sqrt{81 + 289} = \sqrt{370} \approx 19.2 \] So, the distance for Rescue Boat 2 is approximately 19.2 units.

Step 5: Compare Distances

Now, we compare the two calculated distances: Rescue Boat 1's distance is approximately 21.1 units and Rescue Boat 2's distance is approximately 19.2 units. Since 19.2 is less than 21.1, Rescue Boat 2 is closer to the distressed boat.

Key Concepts

Coordinate Geometry

In coordinate geometry, we define locations using a coordinate system, typically a 2D plane with an x-axis and a y-axis. Each point on this plane can be represented by a pair of values called coordinates, usually written as \(x, y\). These values indicate how far along the horizontal x-axis and the vertical y-axis the point is located.
In this context, coordinate geometry allows us to plot the positions of various objects, such as the rescue boats and the distressed craft, on a plane. Understanding the positions of these points is the first step in solving problems related to distances or paths between them.

• The first number in the pair (x-coordinate) determines the placement along the horizontal axis.
• The second number (y-coordinate) determines the placement along the vertical axis.

Coordinate geometry is essential in various real-world applications, including navigation, where precise location tracking is crucial.

Distance Calculation

Distance calculation in a coordinate plane involves finding the length between two points. The distance formula, derived from the Pythagorean theorem, is a critical tool in this process. It allows us to determine how far apart two points are in a straightforward manner.
The distance formula is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here’s how it works:

• Subtract the x-coordinates of the two points to find the difference along the x-axis.
• Subtract the y-coordinates of the two points to find the difference along the y-axis.
• Square both differences; this step ensures all values are positive and consider the full extent of the change.
• Add these squared differences together.
• Take the square root of the sum to find the distance.

The formula simplifies complex spatial relationships into manageable calculations, especially useful in rescue operations or navigation where time and precision are vital.

Problem Solving Steps

Effective problem-solving in mathematics or practical applications involves following clear, logical steps. For the given exercise, applying the distance formula is crucial to find the shortest path. Here is a streamlined approach:
First, understand the problem by identifying what is being asked. In this scenario, we needed to determine which rescue boat would reach a distressed boat quickest. Identifying the starting points and destination is essential in understanding the scope.

• Use the given coordinates to plot the problem visually if possible, aiding conceptual understanding.
• Calculate the distances using the formula for each option, substituting the coordinates of the boats and the distressed craft into the distance formula.
• Once distances are calculated, compare them to determine the shortest one. This approach helps in deciding efficiently which boat should proceed, saving time and potentially lives in a rescue scenario.

Following structured steps helps efficiently tackle any problem requiring the application of formulae or logical reasoning, enhancing clarity and accuracy.