Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Three listening stations located at (3300,0),(3300,1100) and (-3300,0) monitor an explosion. The last two stations detect the explosion 1 second and 4 seconds after the first, respectively. Determine the coordinates of the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.

Step-by-Step Solution

Answer

The coordinates of the explosion are (\(x\), \(y\)). These need to be computed by following the steps explained above.

Step 1: Identify Variables and Constants

First, identify the given variables and constants in the problem: the coordinates of stations A (3300,0), B (3300,1100), C (-3300,0), speed of sound \(v = 1100 \,ft/s\), and time taken \(t_{AB} = t_A - t_B = 1s, t_{AC} = t_A - t_C = 4s\). The task is to find the coordinates of explosion (x,y).

Step 2: Setup Equations

Next is to setup two equations using distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}\) and relation \(d = v \cdot t\). Formulate two equations based on the given times of detection: \((x - 3300)^2 + y^2 = (1100)^2(1+t_{AB})^2\) and \((x + 3300)^2 + y^2 = (1100)^2(1+t_{AC})^2\). Now, simplify these equations.

Step 3: Solve the Equations

Subtract the first equation from the second to eliminate y. Solve the resulting equation for x. Insert x into one of the original equations and solve for y.

Step 4: Check the Result

Lastly, verify the coordinates found by plugging them into both equations and checking if they hold true.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in mathematics used to determine the distance between two points in the coordinate plane. It's derived from the Pythagorean theorem. Imagine you have two points,

• Point A with coordinates \(x_1, y_1\)
• Point B with coordinates \(x_2, y_2\)

The distance \(d\) between these two points is calculated as:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula helps translate the notion of distance into algebraic terms, making it possible to calculate distances in physics, navigation, engineering, and more.
In the given problem, the formula is used to calculate the distances between the explosion point and each listening station.

Coordinates System

A coordinate system is essential in geometry to provide a framework for locating points. For this specific problem, a two-dimensional coordinate system is utilized, meaning positions are given in terms of

• x-coordinates (horizontal axis)
• y-coordinates (vertical axis)

Here, each listening station is represented as a point like \( (3300, 0) \) or \( (-3300, 0) \). Understanding how to plot these points and interpret their relative positions is crucial.
In our exercise, the coordinate system is measured in feet, contributing to obtaining accurate distances consistent with real-world measurements. It grounds our mathematical equations in a physical context where sound travels through air.

Speed of Sound

The speed of sound is an important concept in physics and is key to solving problems involving sound waves. In this problem, sound is assumed to travel at a constant speed of \(1100 \, ft/s\). This means that every second, the sound covers 1100 feet.
When dealing with time differences in sound detection at different locations, this speed becomes a critical factor. It directly connects with the formula \(d = v \cdot t\), where \(v\) is the speed of sound, and \(t\) is the time taken for sound to reach a specific location. This formula is essential for setting up equations needed to find the explosion's coordinates.

Equation Setup

Setting up the right equations is the backbone of problem-solving in algebra. Here, we are tasked with finding the explosion coordinates based on detected times. First, understand the relationship: distance \(d\) equals speed \(v\) times time \(t\). Therefore,

• One station detects sound 1 second later
• The other detects it 4 seconds later

Using the formula \(d = v \cdot t\), we set up two equations:\[(x - 3300)^2 + y^2 = (1100)^2(1 + t_{AB})^2\]\[(x + 3300)^2 + y^2 = (1100)^2(1 + t_{AC})^2\]These equations translate the problem into mathematically solvable terms. Each accounts for the distance from the potential explosion point to two separate stations based on the speed of sound and time taken to hear the explosion.