Problem 45 - Calculus for Biology and Medicine

Question

Assume that the distance a car travels is proportional to the time it takes to cover the distance. Find an equation that relates distance and time if it takes the car 15 min to travel \(10 \mathrm{mi}\). What is the constant of proportionality if distance is measured in miles and time is measured in hours?

Step-by-Step Solution

Answer

The constant of proportionality is 40, and the equation relating distance and time is \( D = 40T \).

Step 1: Understand the Proportional Relationship

The relationship between distance and time suggests that distance (D) is proportional to time (T), which can be expressed as: \[ D = kT \]where \( k \) is the constant of proportionality.

Step 2: Convert Time to Appropriate Unit

Since the distance is given in miles and we need to find the constant of proportionality in terms of hours, convert 15 minutes to hours:\[ T = \frac{15}{60} = 0.25 \text{ hours} \]

Step 3: Solve for the Proportionality Constant k

Using the distance \( D = 10 \text{ miles} \) and the time \( T = 0.25 \text{ hours} \), substitute these values into the equation:\[ 10 = k \times 0.25 \]Now, solve for \( k \):\[ k = \frac{10}{0.25} = 40 \]

Step 4: Write the Equation Relating Distance and Time

With \( k = 40 \), the equation that relates distance and time is:\[ D = 40T \]

Key Concepts

Distance-Time Relationship

A Distance-Time Relationship is a foundational concept in understanding how objects move. When we say that distance is proportional to time, we are expressing a simple yet powerful idea about movement. Imagine you are on a road trip. The farther you travel, the longer you have been driving. This direct relationship can be described with the equation:

• \( D = kT \) where:
• \( D \) is the distance traveled.
• \( T \) is the time spent traveling.
• \( k \) is the constant of proportionality.

Thus, if you spend more time traveling at a consistent rate, you cover more distance, illustrating a linear relationship. In our exercise, a car travels 10 miles in 15 minutes. This tells us that the car keeps a steady speed, or constant rate, over the entire journey. Always remember, when dealing with a proportional relationship, the distance grows in direct proportion to the time.

Constant of Proportionality

The Constant of Proportionality is the magic number that ties two proportional quantities together. In our distance-time relationship, this constant tells us how fast something is moving. For example, if you know the car travels 10 miles in 15 minutes, you can calculate how many miles it covers per hour. Let's break this down:

• You start with the formula \( D = kT \).
• Given \( D = 10 \text{ miles} \) and \( T = 0.25 \text{ hours} \).
• To find \( k \), solve \( 10 = k \times 0.25 \).
• Dividing both sides by 0.25 gives \( k = 40 \).

This constant, \( k = 40 \), tells us the car travels at a rate of 40 miles per hour (mph). Essentially, in our context, it's the speed of the car. Recognizing the constant of proportionality helps us forecast outcomes, such as predicting how long a journey might take or how far you'll go in a set time.

Unit Conversion

Unit Conversion plays an essential role in solving many mathematical problems, including our exercise. For consistency, it's crucial to use compatible units across all parts of your calculations. Here, the exercise provides a conversion task:The car travels 10 miles in 15 minutes. However, to find the rate as a constant of proportionality, you need time expressed in hours, not minutes. Thus:

• Convert minutes to hours by dividing: \( 15 \text{ minutes} / 60 = 0.25 \text{ hours} \).

Understanding conversions allows you to seamlessly shift between units without changing the inherent relationship of the values. Just like translating a language, converting units helps ensure you can communicate and calculate uniformly across different measurements.By mastering unit conversions, you'll be able to handle a wide variety of real-world problems, ensuring accuracy in your calculations and findings.