Problem 6 - Algebra 2 and Trigonometry

Question

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. On his way to work, Randy travels 10 miles at \(r\) miles per hour for \(h\) hours.

Step-by-Step Solution

Answer

The variables vary inversely.

Step 1: Understand Direct and Inverse Variation

In direct variation, one variable increases as the other increases. The relationship can be described by the formula: \( y = kx \) where \( k \) is a constant. In inverse variation, one variable increases as the other decreases, which is described by the formula: \( y = \frac{k}{x} \). Neither indicates that the variables do not have a simple direct or inverse relationship.

Step 2: Identify Variables in the Problem

In this problem, the key variables involved are the speed \( r \) (in miles per hour) and time \( h \) (in hours). The distance traveled is 10 miles, which is a constant.

Step 3: Relate the Variables with Distance Formula

The relationship between distance, speed, and time is given by the formula: \( ext{Distance} = ext{Speed} imes ext{Time} \). In this problem, it is \( 10 = r imes h \). This implies that \( r imes h = 10 \).

Step 4: Determine the Type of Variation

Since \( r imes h = 10 \), if \( r \) increases, \( h \) must decrease to maintain a constant distance of 10. This describes an inverse relationship because one variable decreases as the other increases. Therefore, \( r = \frac{10}{h} \) and \( h = \frac{10}{r} \), consistent with inverse variation.

Key Concepts

Direct Variation

Direct variation describes a specific type of relationship between two variables where one variable changes in the same direction as the other. If one variable increases, the other also increases, and this is described by a linear equation of the form:

• \( y = kx \)

Here, \( k \) is a constant known as the constant of variation. This equation implies a proportional relationship between \( y \) and \( x \). For example, if you have a car traveling at a constant speed, the distance traveled varies directly with the time spent traveling. When you double the time, the distance also doubles if speed remains constant.
In the context of the problem with Randy traveling to work, if the scenario involved direct variation, increasing the speed \( r \) while keeping time \( h \) constant would mean double the distance traveled in the same period, which does not align with the given scenario.

Distance Formula

The distance formula describes how distance relates to speed and time. This is a foundational concept in physics and is expressed mathematically as:

• \( \text{Distance} = \text{Speed} \times \text{Time} \)

In this equation, the product of speed (usually in miles per hour) and time (in hours) gives the total distance traveled (in miles). Thus, if you know two of these variables, you can always find the third by rearranging the formula.
In the problem, it reads \( 10 = r \times h \), where 10 is the fixed distance Randy travels to work. This equation can help you understand the balance between speed and time required to cover a fixed distance. When speed \( r \) increases, the necessary time \( h \) decreases, ensuring that the distance remains constant, demonstrating the nature of inverse variation in this instance.

The Relationship Between Speed and Time

Understanding the relationship between speed and time is crucial for determining how they interplay to affect distance traveled. In this inverse relationship, if the speed \( r \) is increased, the time \( h \) must decrease to maintain the same distance, and vice versa.

• Increased speed requires less time for the same distance: \( r = \frac{10}{h} \)
• Increased time requires lower speed to keep the distance the same: \( h = \frac{10}{r} \)

This inverse relationship shows that as one factor increases, the other must proportionally decrease to maintain the constant value. This is clear from Randy's commute, where the distance \( r \times h = 10 \) remains unchanged, showing how speed and time need to adjust inversely to balance each other.