Problem 103 - College Algebra with Modeling and Visualization

Question

Distance to Lightning Find a formula for a function \(f\) that computes the distance between an observer and a lightning bolt when the speed of sound is 1150 feet per second. Evaluate \(f(15)\) and interpret the result.

Step-by-Step Solution

Answer

The function is \( f(t) = 1150t \), and \( f(15) = 17250 \), meaning the observer is 17,250 feet away.

Step 1: Understand the Problem

We need to find a formula that calculates the distance (in feet) from an observer to a lightning bolt based on the time (in seconds) it takes for the sound of thunder to reach the observer. The speed of sound given is 1150 feet per second.

Step 2: Identify the Variables

Let the variable \( t \) represent the time in seconds from seeing the lightning bolt to hearing the thunder. We need to establish a function \( f(t) \) where \( f(t) \) is the distance in feet.

Step 3: Derive the Formula

We can use the speed of sound to calculate the distance with the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]. Therefore, \( f(t) = 1150 \times t \).

Step 4: Evaluate the Function for f(15)

Substitute \( t = 15 \) seconds into the function \( f(t) = 1150 \times t \): \[ f(15) = 1150 \times 15 = 17250 \] feet.

Step 5: Interpret the Result

The result \( f(15) = 17250 \) means that if it takes 15 seconds for the sound of thunder to reach the observer after seeing the lightning bolt, the observer is 17,250 feet away from the lightning bolt.

Key Concepts

Distance Formula

The distance formula is a key mathematical tool used to determine how far apart two points are. In our context, it helps calculate the distance between an observer and a lightning bolt based on the time it takes for the sound to travel. The formula is quite simple:

• Distance = Speed × Time.

Here, the speed is a constant, which is the speed of sound. Knowing the speed and the time the sound takes to travel to the observer, you can multiply these two values to find the distance.

In the given exercise, the speed of sound is 1150 feet per second. So, if it takes 15 seconds after the lightning for the thunder to reach you, you just multiply 1150 by 15 to find how far the lightning struck: 17,250 feet. This formula is straightforward and a handy way to quickly grasp distances in such scenarios.

Functions

Functions are mathematical statements that describe a relationship between inputs and outputs. They are essential building blocks in algebra and provide ways to model real-world phenomena.
In terms of lightning and thunderstorms, a function can be created to express the relationship between the time after a lightning strike (input) and the distance from that strike (output).

• For example, our function is noted as \( f(t) = 1150 \times t \), where \( t \) is time in seconds.

This notation is concise and powerful because it allows you to plug in different times \( t \) to compute various distances. Functions simplify complex situations and enable calculations that are both replicable and scalable.

Using functions well allows for accurate and swift problem-solving in both academic settings and real life.

Speed of Sound

The speed of sound is an important concept when discussing phenomena like thunderstorms because it governs how quickly sound waves travel through the air.
At sea level, the speed of sound is approximately 1150 feet per second. However, this can vary depending on factors like air temperature and pressure.

• In our problem, this constant speed allows us to compute the distance from the lightning by observing the delay between the flash and the resulting thunder.

It's fascinating how this consistent speed helps us unlock information about our environment. Knowing that sound travels at a predictable pace forms the basis for practical applications like measuring distances, locating phenomena, and even synchronizing events.

Paying attention to such constants can greatly enhance your understanding of physical events and their underlying mechanics.