Problem 52 - Precalculus with Limits

Question

SOUND LOCATION You and a friend live 4 miles apart (on the same "east-west" street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.)

Step-by-Step Solution

Answer

The quadratic equation \((21120 - x)/1100 - x/1100 = 18\) is the equation that gives the possible places where the lightning could have occurred.

Step 1: Convert Miles to Feet

Since all the other measurements are given in feet and seconds, the distance between the two observers should also be converted into feet. There are 5280 feet in a mile, so the distance is \(4 miles * 5280 feet/mile = 21120 feet\).

Step 2: Set Up the Equations

Using the speed equals distance over time equation, \(s = d/t\), it can be set up two equations based on the observers' locations. Let x be the distance from the lightning strike to you, and let \(d - x\) be the distance from the lightning to your friend. The time it takes for sound to reach you is \(x/1100\) seconds and the time to reach your friend is \((d - x)/1100\) seconds.

Step 3: Formulate the Equation

The difference in time of you and your friend hearing the thunder is 18 seconds. Thus, \((d - x)/1100 - x/1100 = 18\), which simplifies to \((21120 - x)/1100 - x/1100 = 18\)

Step 4: Solve for X

You can solve for \(x\) to find the possible locations of the lightning strike. This can be done by multiplying every term by 1100 to eliminate the fraction, then simplifying and solving the resulting quadratic equation. Given the nature of the equation, two solutions can be expected indicating two possible locations for the lightning strike. But only solutions where 0 < x < d are valid, since the lightning can't have occurred beyond the observer's locations.

Key Concepts

Quadratic Equations

Quadratic equations are foundational to algebra and appear frequently in various real-world scenarios, including physics, engineering, and economics. They are polynomial equations of the second degree, which means their highest exponent is two. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are constants, and \( x \) represents the unknown variable.

The solutions to a quadratic equation are known as the 'roots', and they can be found using different methods including factoring, completing the square, using the quadratic formula, or graphically. In the context of our sound location problem, the quadratic equation arises from setting up an equality between the distances sound travels in a given time. By manipulating the setup, we end up with a quadratic equation in the form of \( ax^2 + bx + c = 0\), which needs to be solved to find the possible locations of the lightning strike.

Why Quadratics Matter in the Sound Location Problem

Understanding how to solve quadratic equations is crucial for the sound location problem as it helps us determine the two potential points where the lightning could have struck. It involves setting up the problem correctly and then simplifying the terms to reach a solvable format.

Distance Rate Time

The distance rate time relationship is a critical concept in understanding motion and is described by the formula \( d = rt \), where \( d \) is distance, \( r \) is rate (or speed), and \( t \) is time. This equation is a straightforward expression of how the three variables are interconnected. In physics, it is particularly used to solve problems involving constant speeds.

In our exercise, this relationship helps us develop equations that represent the distance sound travels from the lightning strike to each observer. Given that the speed of sound is constant at 1100 feet per second, we can work out how far the sound traveled by multiplying the speed by the time it took to reach each person.

Applying Distance Rate Time to the Problem

By using the distance rate time relationship, we are able to set up two separate equations that correspond to each observer's experience of the thunder sound, taking into account the different times it takes for the sound to reach them. The difference of 18 seconds becomes the key in linking these equations and forms the basis of the quadratic equation to solve.

Unit Conversion

Unit conversion is the process of converting the value of a physical quantity from one unit to another. It's essential in science and engineering to ensure that all quantities are expressed in the same units before performing calculations.

In our sound location problem, the conversion of distances from miles to feet is necessary because speed and time are given in feet per second and seconds, respectively. Failing to convert units could lead to incorrect calculations and results. The step-by-step solution converts the distance between you and your friend into feet using the conversion fact that there are 5280 feet in a mile.

The Importance of Accurate Unit Conversion

Unit conversion is a fundamental skill that must be applied correctly. In the context of this exercise, it is crucial to convert the four-mile distance to 21120 feet because working within a single, consistent unit system is the only way to uphold the accuracy of the distance, rate, and time calculations.