Problem 16 - Intermediate Algebra

Question

Tell whether each relationship suggests direct or inverse variation. Lightning. The time it takes you to hear the lightning after a strike and your distance from the strike

Step-by-Step Solution

Answer

It suggests a direct variation.

Step 1: Identify the Variables

Let's identify the two variables in the situation: the time it takes to hear the thunder (\( t \)) and the distance from the strike (\( d \)).

Step 2: Understand Direct Variation

In direct variation, two variables increase or decrease together, meaning if one variable gets larger, the other does too. The relationship can be expressed as \( y = kx \) where \( k \) is a positive constant.

Step 3: Understand Inverse Variation

In inverse variation, one variable increases as the other decreases, meaning if one variable gets larger, the other gets smaller. The relationship can be expressed as \( y = \frac{k}{x} \) where \( k \) is a constant.

Step 4: Analyze the Relationship

Consider the scenario: The further you are from the lightning (greater \( d \)), the longer it takes to hear the sound (greater \( t \)). Thus, both variables increase together.

Step 5: Conclusion

Since as the distance increases, the time also increases, this suggests a direct relationship between time and distance.

Key Concepts

Variables in Algebra

Variables are symbols used to represent numbers in mathematical equations and expressions. In algebra, variables play a crucial role in forming relationships between different elements. In the context of algebraic expressions, variables often represent quantities that can change or vary. For instance, in the equation \( y = mx + b \), \( y \) and \( x \) are variables, while \( m \) and \( b \) are constants.

• Variables can stand in for unknowns that you need to solve.
• They can also represent real-world quantities such as time, distance, or speed.

In our exercise, the variables are the time \( t \) it takes to hear lightning and the distance \( d \) from the strike. Understanding these variables allows us to explore how they relate to each other and determine whether the relationship is one of direct or inverse variation.

Direct Variation

Direct variation describes a relationship where two variables change in the same manner; meaning, when one variable increases, the other also increases proportionally, and the same applies when they decrease. This kind of relationship can be written mathematically as \( y = kx \), where \( y \) varies directly with \( x \) and \( k \) is a constant factor known as the constant of variation.

• This constant remains the same even as the values of \( y \) and \( x \) change, preserving the relationship.
• A graph of direct variation is a straight line that passes through the origin (0,0).

In the scenario with lightning, as you increase your distance \( d \) from the lightning strike, the time \( t \) it takes to hear it also increases. Because both variables move in the same direction (both increase), this situation suggests a direct variation between time and distance.

Inverse Variation

Inverse variation occurs when one variable increases while the other decreases. This relationship can be captured by the formula \( y = \frac{k}{x} \), where \( y \) varies inversely with \( x \) and \( k \) is a constant. As one variable becomes larger, the reciprocal ensures that the other diminishes.

• The constant \( k \) ensures a predictable change in one variable as the other changes.
• A graph of inverse variation appears as a hyperbola, as the curve shows one variable decreasing while the other increases.

Unlike direct variation, inverse variation would mean in our exercise with lightning, if the time to hear the strike was to decrease as the distance increased, which is not the case here. Therefore, the relationship cannot be described by inverse variation based on our observation of the variables.