Problem 54 - Elementary and Intermediate Algebra

Question

Give examples of two quantities from everyday life that vary directly and two quantities that vary inversely.

Step-by-Step Solution

Answer

Direct variation: distance and time at constant speed, circumference and diameter. Inverse variation: speed and time for constant distance, workers and time for a job.

Step 1: Understanding Direct Variation

Direct variation occurs when two quantities increase or decrease together, maintaining a constant ratio. Mathematically, if quantity \( y \) varies directly with \( x \), this relationship can be expressed as \( y = kx \), where \( k \) is the constant of proportionality.

Step 2: Example 1 of Direct Variation

A classic example of direct variation is the relationship between distance and time when driving at a constant speed. If speed \( s \) is constant, the distance \( d \) travelled is directly proportional to time \( t \). Thus, \( d = st \). Here, \( s \) is the constant of proportionality.

Step 3: Example 2 of Direct Variation

The relationship between the circumference and the diameter of a circle is another example of direct variation. The formula is \( C = pi D \), where \( C \) is the circumference, \( D \) is the diameter, and \( pi \) (approximately 3.14159) is the constant of proportionality.

Step 4: Understanding Inverse Variation

Inverse variation occurs when one quantity increases while the other decreases, and the product of the two quantities is constant. Mathematically, if quantity \( y \) varies inversely with \( x \), the relationship is expressed as \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.

Step 5: Example 1 of Inverse Variation

One example of inverse variation is the relationship between speed and travel time for a constant distance. For a given distance, the travel time \( t \) is inversely proportional to speed \( s \). Mathematically, \( t = \frac{d}{s} \), where \( d \) is the constant distance.

Step 6: Example 2 of Inverse Variation

Another example is the relationship between the number of workers and the time required to complete a job. If more workers are added to finish a task, the time \( t \) to complete the job is inversely proportional to the number of workers \( n \), assuming they work at the same rate. This is expressed as \( t = \frac{k}{n} \), where \( k \) is the total amount of work to be done.

Key Concepts

Direct Variation

In our daily experiences, there are situations where two quantities grow together in harmony. When this happens, we say these quantities exhibit direct variation. In mathematical terms, direct variation means that if quantity \( y \) varies directly with quantity \( x \), you can express this relationship with the formula \( y = kx \), where \( k \) is the constant of proportionality.
Think of this \( k \) as the glue that keeps their ratio constant.
Here are some real-world examples to make it clearer:

• Picture yourself on a road trip, driving steadily at a fixed speed. If your car maintains a constant speed, the distance traveled \( d \) is directly proportional to the time spent driving \( t \). The formula \( d = st \) perfectly illustrates this direct relationship, where \( s \) (speed) is the constant of proportionality.
• Another everyday example is the relationship between the circumference of a circle and its diameter. The formula \( C = \pi D \) shows direct variation, with \( \pi \) being approximately 3.14159, acting as our constant \( k \).

These examples highlight how direct variation simplifies complex relationships, making them predictable and easy to understand.

Inverse Variation

Sometimes, in our daily routine, we come across situations where one quantity goes up while another one goes down simultaneously. This phenomenon is known as inverse variation. Mathematically, when quantity \( y \) varies inversely with quantity \( x \), it can be expressed by the formula \( y = \frac{k}{x} \). Here, \( k \) is the constant of proportionality, representing a fixed product.
Here are some common examples:

• Think about traveling a set distance. If the speed \( s \) increases, the time \( t \) it takes to cover that distance decreases. This relationship is described by the formula \( t = \frac{d}{s} \), where \( d \) is the constant distance—the consistent product in this inverse variation scenario.
• Another example is in the workplace. Imagine a job needs to be finished. As more workers \( n \) are added, the time \( t \) needed to complete the job reduces, provided everyone works at the same rate. This is shown by \( t = \frac{k}{n} \), where \( k \) represents the total work needed.

Understanding inverse variation helps us recognize patterns where one element's rise leads to another's fall, allowing us to anticipate outcomes in various practical situations.

Constant of Proportionality

The constant of proportionality is a crucial piece in the puzzle of understanding both direct and inverse variations. It's often denoted as \( k \) and acts as a bridge connecting the varying quantities with a consistent ratio or product.
In the case of direct variation, \( k \) ensures that the ratio of the two quantities remains unchanged. For instance, if you double the time spent traveling at a constant speed, the distance traveled also doubles, maintaining that steady ratio governed by \( k \).When it comes to inverse variation, \( k \) holds the product of the two varying quantities constant. This means if one quantity increases, the other decreases in such a way that their product around \( k \) stays steadfast. For example, if you shorten the time available for completing a job without altering the total amount of work, more workers are required—a balancing act facilitated by \( k \).Understanding the constant of proportionality is vital because:

• It helps in predicting and calculating other related variables effectively.
• It provides a clear depiction of how relationships between quantities are structured in both direct and inverse cases.

The constant of proportionality is the backbone of variation relationships, ensuring that they are easy to manage and comprehend.