Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Distance between Two Factories Two factories blow their whistles at exactly \(5: 00 .\) A man hears the two blasts at 3 seconds and 6 seconds after \(5: 00,\) respectively. The angle between his lines of sight to the two factories is \(42.20^{\circ} .\) If sound travels 344 meters per second, how far apart are the factories to the nearest meter?

Step-by-Step Solution

Answer

The factories are approximately 1877 meters apart.

Step 1: Analyze the Sound Travel Time

First, determine the distances of the man from each factory based on the time it takes for the sound to reach him. The sound from the first factory takes 3 seconds and from the second, 6 seconds.

Step 2: Calculate Distances from the Man to Each Factory

Using the speed of sound, which is 344 meters per second, calculate: - Distance from the first factory: \(344 \times 3 = 1032\) meters. - Distance from the second factory: \(344 \times 6 = 2064\) meters.

Step 3: Apply the Law of Cosines to Find the Distance Between Factories

With the man's two distances and the angle between his lines of sight, use the Law of Cosines to find the distance between the factories:\[ c^2 = a^2 + b^2 - 2ab \cos(C)\]where:- \(c\) is the distance between factories,- \(a = 1032\) meters,- \(b = 2064\) meters,- \(C = 42.20^{\circ}\)Substitute the values:\[c^2 = 1032^2 + 2064^2 - 2 \times 1032 \times 2064 \times \cos(42.20^{\circ})\]

Step 4: Solve for the Distance Between Factories

Calculate:\[c^2 = 1032^2 + 2064^2 - 2 \times 1032 \times 2064 \times \cos(42.20^{\circ}) \approx 3542400.4\]Taking the square root gives:\[c \approx \sqrt{3542400.4} \approx 1877\] meters.

Key Concepts

Trigonometry

Trigonometry is a field of mathematics that studies the relationships between angles and sides of triangles. It's essential in solving problems where you deal with triangles, such as the one in this exercise. Here, we use the **Law of Cosines**, a fundamental theorem in trigonometry, especially useful when we don't deal with right-angled triangles. This law helps find the length of a side when you know:

• The lengths of the other two sides
• The angle between those sides

This theorem is written as:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]In our solution, we use this formula to find the distance between the two factories, considering the man who hears the sound is at the third point of a triangle. Understanding these angles and lengths is crucial. Trigonometry reveals how these are intricately connected, allowing precise calculations.

Distance Calculation

Distance calculation often involves using speed, time, and in some cases, angles. In our example, we determine how far each factory is from the man hearing the sound, using **distance = speed × time**. Knowing the speed of sound allows us to quickly calculate:

• The distance from the first factory as 1032 meters (344 m/s × 3 seconds)
• The second factory as 2064 meters (344 m/s × 6 seconds)

These straight-line distances are vital to forming the sides of our triangular setup. Once these distances are clear, and we know the angle between the factories from the man's perspective, we move on using trigonometry (specifically the Law of Cosines) to solve the final distance between the two factories. Accurate distance calculation is often the first step in order to proceed to more complex trigonometric calculations.

Sound Speed

Sound speed is the velocity at which sound waves travel through a medium, like air. On average, sound speed is about 344 meters per second in air at room temperature. This constant speed means you can calculate how far sound travels over time. In the exercise, sound takes 3 and 6 seconds to travel from each respective factory to the man, which directly relays the distance from him to each source using sound speed.
This concept is pivotal here, as it bridges the gap between time (how long it takes to hear the factory whistles) and distance (how far away those factories are). Without knowing the speed of sound, we couldn't translate seconds into meters, the unit required to solve our distance between the factories using trigonometry. Understanding sound speed is crucial when working with problems involving sound travel and related calculations.