Problem 68 - Introductory Algebra for College Students

Question

The formula $$M=\frac{n}{5}$$ models your distance, \(M,\) in miles, from a lightning strike in a thunderstorm if it takes \(n\) seconds to hear thunder after seeing the lightning. Use this formula to solve. If you are 3 miles away from the lightning flash, how long will it take the sound of thunder to reach you?

Step-by-Step Solution

Answer

By substituting the values into the formula, it's found that it will take 15 seconds for the sound of thunder to reach if you are 3 miles away from the lightning flash.

Step 1: Understanding the problem

The provided formula allows to calculate the time it takes for the sound of thunder to reach a particular point after a lightning flash, given the distance from the flash. The distance \(M\) is given as 3 miles.

Step 2: Rearranging the formula to solve for n

The formula can be rearranged to calculate \(n\) (the time in seconds to hear the thunder) given \(M\) (the distance in miles). The rearranged formula is \(n = 5 \cdot M\).

Step 3: Substitute the given value into the formula

Substitute the given value \(M = 3\) miles into the formula to find out the value of \(n\). So, \(n = 5 \cdot 3 = 15\) seconds.

Key Concepts

Lightning and Thunder

Lightning is a fascinating natural phenomenon that often occurs during thunderstorms. When a lightning strike happens, it creates a massive electrical discharge through the atmosphere. This release of energy is seen as a bright flash of light.
However, the flash isn't just about light; it also involves sound. The lightning heats the air rapidly, creating a sound wave known as thunder. Interestingly, light travels much faster than sound through the air, which is why we often see the flash before hearing the thunder that follows. This difference in speed is critical in determining how far away a lightning strike is.

• The flash of lightning is almost instantaneous.
• Thunder travels at approximately 343 meters per second.
• This time delay allows us to estimate the distance to the lightning.

Calculating Time for Thunder

Time calculation plays a crucial role in understanding how far away a storm is. In this context, the formula \( M = \frac{n}{5} \) helps calculate the distance \(M\) when the time \(n\) is given. It works because the difference between the speed of light and sound allows measuring the distance based on time.
To determine the time it takes for thunder to reach you if you know the distance, simply rearrange the formula to solve for \(n\). The formula becomes \( n = 5 \times M \). Here, \(M\) is the distance in miles, and multiplying it by 5 gives the time \(n\) in seconds.
For example, if you are 3 miles away from a lightning strike, the time to hear the thunder is calculated as:

• Substitute \(M = 3\) into the equation \( n = 5 \, \times \, 3 \).
• You'll find \( n = 15 \) seconds.

It's a simple yet effective way to gauge the location of a storm.

Understanding Thunderstorm Distance

Estimating the thunderstorm distance is a practical use of the distance formula. When you see lightning but haven't yet heard thunder, you can determine how far the storm is by counting the seconds between them.
Here's how it works:

• Watch for a lightning flash.
• Start counting the seconds until you hear thunder.
• Divide the number of seconds by 5 to find the distance in miles.

This method is handy for quickly assessing a storm's distance without specialized tools. For instance, if 10 seconds pass between the lightning and thunder:

• This means the storm is \( \frac{10}{5} = 2 \) miles away.

This easy calculation can help you stay safe by knowing how far away a storm is.