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; the equation is \(d = 40t\).

Step 1: Identify Given Values

We are given that a car takes 15 minutes to travel 10 miles. However, we need to convert the time into hours to match the requirement. Since 1 hour equals 60 minutes, 15 minutes is \( \frac{15}{60} = 0.25 \) hours.

Step 2: Understand Proportionality

The distance travelled, \( d \), is proportional to the time taken, \( t \). This means we can write the relationship as \( d = k \cdot t \), where \( k \) is the constant of proportionality.

Step 3: Substitute the Known Values

Substitute the known values into the proportionality equation: \( 10 = k \cdot 0.25 \).

Step 4: Solve for the Constant of Proportionality

To find \( k \), divide both sides of the equation by 0.25: \[ k = \frac{10}{0.25} = 40.\]

Step 5: Write the Final Equation

With the constant of proportionality, \( k \), known, the equation that relates distance and time is \( d = 40t \). This equation indicates that for every hour, the car travels 40 miles.

Key Concepts

Constant of Proportionality

Imagine that the speed at which a car travels is always steady. In mathematics, such a relationship between two variables, where one variable is always a constant multiple of the other, is called proportionality. For this car, the distance it travels is directly proportional to the time it takes.

To find out how far the car travels in any given time, we use a special number, called the "constant of proportionality," symbolized by the letter \(k\). In our example, this is calculated by the formula:

• \(d = k \cdot t\)

Here, \(d\) stands for distance, \(t\) stands for time, and \(k\) is what tells us the rate of travel per unit time. Once you know \(k\), you can predict how far the car will go in any amount of time if it keeps traveling at this constant speed.

In our case, \(k\) was found to be 40. This means for every hour, the car travels 40 miles, no matter what.

Distance-Time Relationship

In the world of math and physics, understanding how distance and time relate can tell us a lot about motion. Specifically, this exercise deals with the linear distance-time relationship of a car. The equation that ties these two dimensions together is called a linear equation, which in our case is

• \(d = 40t\)

This equation is particularly useful:

• It's simple: It only uses multiplication, making it easy to calculate by hand or without special tools.
• It's direct: Changing one side of the equation directly changes the other. If you double the time, you double the distance.

So, when we say distance is proportional to time, it means there's a steady increase in distance as time goes on, as long as the speed is constant.

Watching this distance-time relationship will give you insight into the concept of speed, offering a simpler way to see how long travel takes for varying speeds or timeframes.

Unit Conversion

Considering unit conversion can often seem like an extra step, but it is crucial in maintaining the integrity and accuracy of our calculations. In our exercise, we started with time in minutes but needed it in hours to find the constant of proportionality correctly.

• 1 hour equals 60 minutes, so to convert minutes into hours, divide the number of minutes by 60.
• In our scenario, 15 minutes was converted by calculating \(\frac{15}{60} = 0.25 \) hours.

Making sure both time and distance are in consistent units is important for the proportionality formula to work correctly. Without this conversion, our constant of proportionality would be flawed.

Understanding and applying unit conversion helps in various real-world situations, especially when different systems of measurement collide. Knowing how to convert from minutes to hours, or feet to meters, for example, can make calculations in science, engineering, and daily life much easier and more accurate.