Problem 67 - Introductory and Intermediate Algebra for College Students 4th

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 2 miles away from the lightning flash, how long will it take the sound of thunder to reach you?

Step-by-Step Solution

Answer

It will take 10 seconds for the sound of thunder to reach you if you are 2 miles away from the lightning flash.

Step 1 - Substitute the given value

Replace \(M\) in the equation \(M = n/5\) with the provided value which is 2 miles. So the equation becomes \(2 = n/5\).

Step 2 - Solve for \(n\)

The main aim is to isolate \(n\) on one side of the equation. To do this, multiply both sides of the equation by 5. So, \(2 \times 5 = n\).

Step 3 - Calculate the result

Calculate \(2 \times 5\) which is equal to 10. So, \(n = 10\). This means it will take 10 seconds for the sound of thunder to reach you if you are 2 miles away from the lightning flash.

Key Concepts

Algebraic Equation Solving

The process of algebraic equation solving involves finding the value of unknown variables that satisfy the given equation. In the context of the lightning distance calculation exercise, the unknown variable is the time in seconds, represented by n, it takes for the sound of thunder to travel a certain distance after a lightning flash. The given algebraic equation is \( M = \frac{n}{5} \) and it establishes the relationship between the distance M (in miles) and time n (in seconds).

When provided with the distance, in this case, 2 miles, we substitute that value into the equation and solve for n. The substitution results in the equation \( 2 = \frac{n}{5} \) and by applying basic algebraic principles such as multiplication, we isolate n, yielding \( n = 10 \) seconds. Algebraic equation solving is a fundamental skill needed across various fields of mathematics and science, as it enables us to deduce unknown quantities from established relationships.

Distance-Speed-Time Relationship

The distance-speed-time relationship is a foundational concept in physics, often summarized by the formula \( \text{Distance} = \text{Speed} \times \text{Time} \) or D = S × T. This relationship explains how the three variables are interconnected; knowing any two of them allows for the calculation of the third. In the context of our lightning distance exercise, the speed of sound in the air is implicitly used to derive the given formula \( M = \frac{n}{5} \) (where M is the distance in miles, and n is time in seconds).

Since sound travels at approximately one fifth of a mile per second (which is roughly 1,126 feet per second or 343 meters per second at sea level), this factor guides the equation. Understanding how these variables interact with each other is crucial in solving real-world problems involving motion and sound, such as determining how far away a lightning strike occurred based on the time delay between seeing the flash and hearing the thunder.

Substitution Method

The substitution method is an algebraic technique used to solve equations where one variable can be expressed in terms of another. It is particularly useful in systems of equations but can also be applied to single equations with one unknown, as demonstrated in the lightning distance problem.

Initially, we have the formula \( M = \frac{n}{5} \) which relates distance M to the time n. When we’re given the distance value, we substitute it into the equation in place of M. This method allows for a straightforward calculation when an equation is already solved for one variable as it relates to another. It's a critical skill because it not only simplifies equations but also lays the groundwork for more advanced techniques in algebra and calculus. In this way, substitution serves as a stepping stone to more complex problem-solving methods.