Problem 28 - Calculus

Question

Write a formula representing the function. The average velocity, \(v,\) for a trip over a fixed distance, \(d,\) is inversely proportional to the time of travel, \(t\)

Step-by-Step Solution

Answer

The formula is \( v = \frac{d}{t} \).

Step 1: Understanding the problem

The problem states that the average velocity, denoted as \( v \), is inversely proportional to the time of travel, \( t \). Inverse proportionality implies that as one quantity increases, the other decreases. For two variables \( v \) and \( t \) to be inversely proportional, their product remains constant for a given fixed distance, \( d \).

Step 2: Expressing Inverse Proportionality

To express that \( v \) is inversely proportional to \( t \), we use the formula \( v = \frac{k}{t} \), where \( k \) is the proportionality constant that depends on the fixed distance \( d \).

Step 3: Identifying the Proportionality Constant

Since \( v = \frac{k}{t} \) and we are given that the distance \( d \) is fixed, we know that \( v = \frac{d}{t} \). This is because the average velocity is defined as the total distance divided by total time, hence, \( k = d \).

Step 4: Formulating the Equation

By substituting the value of \( k \) in the inverse proportionality formula, we get \( v = \frac{d}{t} \). This equation represents the relationship between average velocity, distance, and time for a trip over a fixed distance.

Key Concepts

Average Velocity

Average velocity is a crucial concept in understanding motion over a distance. It is the measure of how fast something moves on average over a specific interval of time. This is determined by the ratio of the total distance traveled to the total time taken. In the context of the given exercise, we're dealing with a fixed distance, denoted as \(d\), and the average velocity \(v\) is inversely proportional to the time \(t\).

To put it simply, if you know how far you have traveled and how long it took, you can compute your average velocity using the formula:

• \(v = \frac{d}{t}\)

This implies that if you keep the distance constant, increasing the travel time will decrease the average velocity, and vice versa. Understanding this relationship helps in analyzing how time constraints impact speed when distance is unchangeable.

Proportionality Constant

The concept of a proportionality constant is essential in mathematics, particularly when dealing with relationships between two variables. In this context, the proportionality constant is the fixed distance \(d\).

Since the average velocity \(v\) is inversely proportional to the time \(t\) in the formula \(v = \frac{k}{t}\), where \(k\) is the proportionality constant, understanding its role becomes crucial. The task is to find this constant so that the relationship makes sense.

Here, we understand that the formula for average velocity can be modified to express this constant. By rearranging, we have \(v = \frac{d}{t}\), suggesting that:\

• The proportionality constant \(k\) equals \(d\), the fixed distance.
• The constant \(d\) ensures that as time varies, the product of \(v\) and \(t\) remains equivalent to \(d\).

This constant ties the relationship between velocity and time, maintaining consistency irrespective of either variable's variance.

Time of Travel

Time of travel is integral to the concept of motion and velocity. It refers to the duration taken to cover a specific distance. In many real-world problems, understanding how time impacts velocity is key, especially in inverse proportional scenarios.

Since the average velocity \(v\) is inversely proportional to \(t\), it can be analyzed that:

• When time \(t\) increases, the average velocity \(v\) decreases, provided the distance \(d\) is constant.
• Conversely, if time \(t\) decreases, \(v\) increases.

This relationship shows that managing time efficiently can affect how quickly a distance is traveled. Being aware of this can help optimize time usage during travels or related scenarios. In our exercise, adjusting the time parameter directly influences the speed, underscoring the dynamic between these two quantities.