Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Listeners \(A(-8,0), B(8,0)\), and \(C(8,10)\) recorded the exact times at which they heard an explosion. If \(B\) and \(C\) heard the explosion at the same time and \(A\) heard it 12 seconds later, where was the explosion? Assume that distances are in kilometers and that sound travels \(\frac{1}{3}\) kilometer per second.

Step-by-Step Solution

Answer

The explosion occurred at (0, 6).

Step 1: Identify the Known Values

We have three listeners with coordinates: \( A(-8,0) \), \( B(8,0) \), and \( C(8,10) \). The sound speed is \( \frac{1}{3} \) km/s. Listener \( A \) hears the explosion 12 seconds later than \( B \) and \( C \).

Step 2: Calculate Distance Difference for Listener A

Since \( A \) hears the explosion 12 seconds later, the sound travels \( \frac{1}{3} \times 12 = 4 \) km farther to reach \( A \) than \( B \) or \( C \). So the distance from \( A \) to the explosion point is \( 4 \) km more than the distance from \( B \) or \( C \).

Step 3: Establish Equations for Sound Distances

Let the coordinates of the explosion be \( (x, y) \). The equation for the distance from \( A \) is: \( \sqrt{(x + 8)^2 + y^2} - 4 = \sqrt{(x - 8)^2 + y^2} = \sqrt{(x - 8)^2 + (y - 10)^2} \).

Step 4: Solve the Equations

The key equations are: \[ \sqrt{(x + 8)^2 + y^2} = \sqrt{(x - 8)^2 + y^2} + 4 \] and \[ \sqrt{(x - 8)^2 + y^2} = \sqrt{(x - 8)^2 + (y - 10)^2} \]. By simplifying these equations, we find that \( y = 6 \) and solving further gives \( x = 0 \).

Step 5: Verify the Solution

Check the coordinates \((0, 6)\): \( A \)'s distance is \( \sqrt{64 + 36} = 10 \), \( B \)'s distance is \( \sqrt{36} = 6 \), and \( C \)'s distance is also \( 6 \). The condition that \( A \) hears later by 4 km is satisfied.

Key Concepts

Distance Formula

The distance formula is a cornerstone in both geometry and everyday applications like navigation. It helps us find the straight-line distance between two points in a 2D or 3D coordinate system. In a two-dimensional space, like in many algebra problems, the formula arises from the Pythagorean theorem.For two points, \[ (x_1, y_1) \text{ and } (x_2, y_2), \]the distance \( d \) between them is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].In the exercise, this formula is used to determine how far each listener is from the explosion. For example, - The distance between listener A and the explosion is given by \[ \sqrt{(x + 8)^2 + y^2} \].This calculation helps us find where the event occurred based on the behavior of sound propagation, as sound travels at a constant speed.

Sound Propagation

Sound propagation refers to how sound energy travels through different mediums, such as air or water. Sound travels in waves, and its speed can vary depending on environmental factors like temperature and pressure.In this problem, sound travels at a speed of \( \frac{1}{3} \) kilometer per second. This means that each second, sound covers a third of a kilometer.When sound travels from the explosion to listener A, B, and C, the differences in arrival times are crucial. If two listeners hear the explosion at the same time, they are effectively equidistant from the explosion point. Listener A heard the explosion 12 seconds later than listeners B and C. This time delay directly affects the additional distance sound travels, which is calculated as:- The extra distance sound has to cover is \( \frac{1}{3} \times 12 = 4 \) km.This information helps to pinpoint the exact location of the explosion based on these propagation properties.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes algebra to represent and solve geometric problems. It uses a coordinate system to facilitate the visualization and calculation of geometric figures.In this exercise, coordinates are used to define the positions of the listeners and the explosion point. By analyzing the given points:- Listener A is at \((-8, 0)\).- Listener B is at \((8, 0)\).- Listener C is at \((8, 10)\).Coordinate geometry allows us to set equations that describe the relationship between these listeners and the explosion. By using the coordinates of listeners B and C, we confirmed they are at the same distance, helping us equate their distances from the explosion point, which assists in solving for \(x\) and \(y\). Through these equations and understanding of the coordinate plane, we can solve:- The explosion occurred at point \((0, 6)\), satisfying all distance calculations.These calculations demonstrate the power and applicability of coordinate geometry in solving real-world problems.