Problem 62 - College Algebra

Question

A small craft in Lake 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 is 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: Identify the Problem

We need to determine which of the two rescue boats will reach the distressed boat fastest. This involves finding the distances between the distressed boat and each rescue boat.

Step 2: Recall the Distance Formula

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) can be calculated using the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 3: Calculate Distance from Rescue Boat 1

The coordinates for the distressed boat are (49, 64) and for Rescue Boat 1 are (60, 82). Apply the distance formula: \[d_1 = \sqrt{(60 - 49)^2 + (82 - 64)^2} = \sqrt{11^2 + 18^2} = \sqrt{121 + 324} = \sqrt{445}\]

Step 4: Calculate Distance from Rescue Boat 2

The coordinates for the distressed boat are (49, 64) and for Rescue Boat 2 are (58, 47). Apply the distance formula: \[d_2 = \sqrt{(58 - 49)^2 + (47 - 64)^2} = \sqrt{9^2 + (-17)^2} = \sqrt{81 + 289} = \sqrt{370}\]

Step 5: Compare Distances

Compare the distances \(\sqrt{445}\) for Rescue Boat 1 and \(\sqrt{370}\) for Rescue Boat 2. Since \(\sqrt{370} < \sqrt{445}\), Rescue Boat 2 is closer to the distressed boat.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry through the use of a coordinate system. By representing geometric shapes in a numeric coordinate system, problems become more manageable and calculations can easily demonstrate spatial relationships.

In this exercise, we utilized a coordinate system to locate the positions of a distressed boat and two rescue boats. Each point on a plane is defined by an ordered pair \(x, y\) to denote horizontal and vertical positioning. This method allows for precise calculation of distances, important in scenarios like plotting rescue missions.

• The distressed boat is at coordinates \(49, 64\).
• Rescue Boat 1 is at \(60, 82\).
• Rescue Boat 2 is at \(58, 47\).

Understanding coordinate geometry is essential for problem-solving in diverse fields such as engineering, navigation, and physics.

Triangulation Methods

Triangulation is a method used to determine the location of a point by forming triangles to it from known points. This approach often involves measuring angles and distances and is commonly employed in navigation, surveying, and rescue missions.

In our exercise, although we didn't directly form triangles, understanding distances and their calculations play a vital role in establishing the quickest route to the distressed boat, analogous to considering various triangle-based pathways.

• Distance formula acts similarly to calculating sides of a triangle.
• Despite not using angles here, distance itself is a significant consideration in triangulation methods.

While our immediate concern is measuring line distances, knowing triangulation enhances our appreciation of such scenarios, especially when determining proximity and paths in 2D space.

Problem Solving Strategy

Employing a structured problem-solving strategy like the one here is critical, especially when quick decisions are vital. Mapping out your approach step-by-step not only prevents errors but also enhances understanding and efficiency. Here’s how it was applied:

First, identify the problem that needs solving – which rescue boat will reach the distressed boat first. Next, use the distance formula to calculate and compare each rescue boat's distance to the distressed boat.

• Recognize the problem: Understand the task clearly, breaking it down if needed.
• Recall relevant formulas: The distance formula is key, so ensure familiarity with it.
• Carry out calculations: Perform the necessary calculations diligently, double-checking work to avoid mistakes.
• Compare results: With the calculated distances, a simple comparison yields the answer.

Following this logical process enables accurate and swift decision-making, which is crucial in rescue operations and many real-world scenarios.