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 Los Angeles and Las Vegas is \(434 \mathrm{~km}\). How many miles is this?

Step-by-Step Solution

Answer

The equation is \( y = 1.609x \). The distance from Los Angeles to Las Vegas is approximately 269.60 miles.

Step 1: Understand the Conversion Factor

The conversion factor between miles and kilometers is given as 1 mile equals 1.609 kilometers.

Step 2: Writing the Equation

Since the problem assumes a linear relationship between miles and kilometers, and using the conversion factor, we can express the equation as \( y = 1.609x \), where \( y \) is the distance in kilometers and \( x \) is the distance in miles.

Step 3: Rearrange for Miles

To find miles from kilometers, rearrange the equation \( y = 1.609x \) to solve for \( x \). Thus, \( x = \frac{y}{1.609} \).

Step 4: Plug in the Kilometers

Use the equation derived in the previous step to find the number of miles given kilometers. Substitute \( y = 434 \) km into the formula: \( x = \frac{434}{1.609} \).

Step 5: Calculate the Miles

Perform the division to find \( x \). \( x = \frac{434}{1.609} \approx 269.60 \) miles.

Key Concepts

Miles to Kilometers

Converting miles to kilometers is a common necessity, especially in countries where different systems of measurement are used. This conversion involves a simple multiplication due to the consistent relationship between the two units. One mile is approximately equal to 1.609 kilometers. Therefore, to convert miles to kilometers, you simply multiply the number of miles by 1.609.

Here's how it works:

• Start with your distance in miles.
• Multiply this distance by the conversion factor, 1.609.
• The result is the distance in kilometers.

This straightforward conversion is useful in everyday scenarios like mapping distances in international travel or comparing data in sports and science. Remembering this conversion factor can save time and avoid the need for frequent calculation tools.

Linear Equations

Linear equations represent relationships where the change in one variable is proportional to the change in another. In mathematics, they are often used to express conversions or direct variations, such as in this exercise where the relationship between miles and kilometers is linear given the constant conversion factor.

The general form of a linear equation is written as:

• \( y = mx + b \)
• In this case, \( m \) is the slope, or the conversion factor (1.609),
• \( b \) is the y-intercept, which is zero in this specific conversion, meaning no additional constant is added.

The equation \( y = 1.609x \) directly translates distances in miles (\( x \)) to kilometers (\( y \)), with no offset. Understanding this concept helps in converting and reshaping equations to suit different mathematical or real-world applications.

Distance Measurement

Distance measurement is an essential concept in various fields such as geography, transportation, and physics. It provides a way to quantify how far apart two points are. In this exercise, we looked at measuring distances in both miles and kilometers.

Knowing how to convert between these units is crucial. Here's why:

• Global standardization: Allows consistency in communication internationally.
• Practical applications: Useful in planning travel routes and logistics.

To convert from kilometers to miles, you need to reverse the process we used earlier. Use the equation \( x = \frac{y}{1.609} \). By rearranging the linear equation \( y = 1.609x \), this formula helps us find how many miles correspond to a given number of kilometers, just as it did for the Los Angeles to Las Vegas distance example. Accurately measuring and converting distances is vital for both practical navigation and on a larger scale, for understanding geographic relationships around the globe.