Problem 50 - Calculus for Biology and Medicine

Question

To convert a distance measured in miles to a distance measured in kilometers, we use the fact that 1 mile equals \(1.609\) kilometers. Denote by \(x\) the distance measured in miles and by \(y\) the distance measured in kilometers. Assume a linear relationship between these two units of measurements. (a) Find an equation relating \(x\) and \(y\). (b) The distance between Minneapolis and Madison is 261 miles. How many kilometers is this?

Step-by-Step Solution

Answer

The equation is \(y = 1.609x\). The distance is approximately 419.349 kilometers.

Step 1: Understand the Conversion Relationship

We know that 1 mile is equivalent to \(1.609\) kilometers. Given this, we can express the conversion as a linear equation.

Step 2: Formulate the Equation

Since 1 mile is equal to \(1.609\) kilometers, we can write the equation relating miles and kilometers as \(y = 1.609x\), where \(x\) is the distance in miles and \(y\) is the distance in kilometers.

Step 3: Convert the Given Distance

We need to find out how many kilometers correspond to 261 miles. Using the equation \(y = 1.609x\), substitute \(x = 261\) to get \(y = 1.609 \times 261\).

Step 4: Calculate the Resulting Kilometers

Perform the multiplication: \(y = 1.609 \times 261 = 419.349\). Thus, the distance in kilometers is approximately \(419.349\) km.

Key Concepts

Linear Relationship

In mathematics, a linear relationship between two variables implies that they change in a proportional manner. This means if one variable changes, the other changes by a constant factor. Linear relationships are characterized by equations that produce straight lines when graphed.

For distance conversion, the relationship between miles and kilometers is linear. Why? Because for every mile, there is always the same increase of 1.609 kilometers. The relationship can therefore be expressed in a simple equation: if \(y\) represents kilometers and \(x\) represents miles, the linear equation is \(y = 1.609x\).

• The factor \(1.609\) is the constant of proportionality.
• This equation allows for easy conversion between miles and kilometers.

Understanding this concept helps in quickly converting distances without needing repetitive calculations.

Distance Conversion

Distance conversion is vital for understanding and navigating between various measurement systems. Converting distances involves using a specific formula, so a consistent multiplication factor brings the conversion to life.

Here's how you can approach distance conversion:

• Know the conversion factor: for miles and kilometers, it's the number \(1.609\).
• Set up the conversion equation: \(y = 1.609x\) where \(y\) is kilometers and \(x\) is miles.
• Input the miles value into the equation to find out the equivalent kilometers.

This method makes conversions straightforward and manageable. For example, to convert 261 miles to kilometers, use the formula \(y = 1.609 \times 261\). You will find that it equals approximately 419.349 kilometers.

Thus, converting distances isn't just helpful; it's crucial when traveling or analyzing data across different regions.

Miles to Kilometers

The conversion of miles to kilometers is essential, especially in countries using the metric system. Since the United States predominantly uses miles, understanding this conversion is particularly useful.

Using a fixed conversion factor helps in bridging the gap between these two units.

• Start with the conversion factor: 1 mile = 1.609 kilometers.
• The conversion formula \(y = 1.609x\) ensures you get the correct number of kilometers for any given number of miles.

For instance, to find out how many kilometers are equivalent to the distance from Minneapolis to Madison—which is 261 miles—you compute \(y = 1.609 \times 261\). This results in approximately 419.349 kilometers.

Mastering this conversion not only enriches your mathematical skills but also enhances your understanding of geographical contexts and international travel.