Problem 12 - Intermediate Algebra

Question

The algebraic expression \(\frac{x}{5}\) can be used to calculate the distance in miles that you are from a flash of lightning, where \(x\) is the number of seconds between the time you see a flash of lightning and the time you hear the thunder. Calculate the distance that you are from the flash of lightning if you hear the thunder 2 seconds after you see the lightning.

Step-by-Step Solution

Answer

You are 0.4 miles from the flash of lightning.

Step 1: Identify the variable

In the expression given, \(x\) represents the number of seconds between seeing the lightning and hearing the thunder. Here, \(x = 2\) seconds, as stated in the problem.

Step 2: Substitute the variable

Substitute \(x = 2\) seconds into the algebraic expression \(\frac{x}{5}\). This changes the expression to \(\frac{2}{5}\).

Step 3: Compute the expression

Now, calculate \(\frac{2}{5}\). This is done by dividing 2 by 5, which equals 0.4.

Step 4: Interpret the result

The calculation shows that the distance from the lightning flash is 0.4 miles.

Key Concepts

Distance Calculation

Calculating distance in algebra involves understanding the relationship between variables in an expression. In this exercise, we are dealing with the concept of measuring distance based on the time delay between seeing lightning and hearing thunder. This natural phenomenon allows us to estimate how far lightning has struck using an algebraic expression. Here, we explore how \[\text{Distance} = \frac{x}{5}\]works, where the expression \( \frac{x}{5} \) is suitable for calculating distance in miles. The variable \( x \) represents the seconds elapsed, and you divide it by 5 to convert the time delay into miles.
The division is based on the average speed of sound. For every 5 seconds away, sound travels approximately 1 mile. This is why dividing the seconds by 5 gives the distance in miles. When you understand this, using algebra to determine distances becomes intuitive.

Variable Substitution

Variable substitution is crucial for solving algebraic expressions. In the current problem, the expression \( \frac{x}{5} \) includes \( x \), which is the number of seconds between the lightning flash and the thunder. To make any calculation, you first determine the value of this variable based on observations you make, such as the time counted between seeing the flash and hearing the sound.
Then, substitute this number into the expression wherever \( x \) appears.

• Identify what \( x \) stands for
• Plug the value into the expression
• Simplify the expression to solve it

With substituting \( x = 2 \), the expression transforms to \( \frac{2}{5} \). This is an essential skill in algebra, as it allows you to replace abstract variables with practical numbers you can then work with.

Interpreting Results

After carrying out calculations in algebra, it's important to interpret results correctly. The final number provides insight into what the mathematical operations reveal about real-world situations. In this case, \( \frac{2}{5} \) means 0.4 miles, indicating your distance from the lightning strike. It's not simply about achieving a numerical outcome, but understanding what that outcome represents.
When you conclude that you are 0.4 miles away from the lightning, you better grasp spatial concepts and how mathematical principles apply to everyday phenomena. This interpretation process aids in deeper learning and makes the numbers more relatable and comprehensible. Consider what the numbers insinuate about safety measures during storms.
This reflection leads to logical and informed decision-making, reinforcing the understanding of algebra within practical contexts.