Problem 38 - Algebra Form and Function

Question

The distance \(M,\) in inches, between two points on a map is proportional to the actual distance \(d\), in miles, between the two corresponding locations. (a) If \(1 / 2\) inch represents 5 miles, find the constant of proportionality and give its units.

Step-by-Step Solution

Answer

Answer: The constant of proportionality is 1/10 inch per mile.

Step 1: Understand the relationship between the two distances

The distance on the map (M) is proportional to the actual distance (d). This means that there is a constant ratio between the two distances, which we will represent with the letter k: M = k * d

Step 2: Use the provided information to set up an equation

In the given example, we know that a distance of 1/2 inch on the map represents 5 miles. Using these values, we can set up an equation: 1/2 = k * 5

Step 3: Solve the equation for the constant of proportionality (k)

To find the value of k, we need to solve the equation 1/2 = k * 5. Divide both sides of the equation by 5: k = (1/2) / 5 This simplifies to: k = 1/10

Step 4: Express the constant of proportionality with units

The constant of proportionality, k, is equal to 1/10. Since k represents the relationship between inches (M) and miles (d), its units are inches per mile. Therefore, the constant of proportionality is 1/10 inch per mile.

Key Concepts

Ratios and Proportions

In the world of mathematics, especially when dealing with distances, understanding ratios and proportions is vital. A ratio is a way to compare two quantities by division. For example, saying "1/2 inch to 5 miles" is expressing a ratio. A proportion, on the other hand, is an equation that states that two ratios are equivalent. When we state that one inch on the map corresponds proportionally to the real world distance, we are discussing a proportion.

Consider this: if a map’s scaling is inaccurate, it may lead to misrepresentations. That is why maintaining a consistent ratio is crucial in creating useful and precise maps. When two distances (map to actual) are said to be proportional, they increase and decrease together at a consistent rate. Thus, identifying the constant of proportionality helps ensure the accuracy of such representations. This understanding is not just limited to maps but is applicable in various fields such as architecture, engineering, and even everyday tasks.

Distance Measurement

Distance measurement on maps is a practical application of math. It's crucial to realize that map distances use scaling. This is a method to represent real-world distances in a more manageable, scaled-down size. Think of it like shrinking down a picture to fit inside a frame while maintaining its true content.

When you have a map scale that says 1 inch equals 10 miles, the scale acts as a conversion tool, transforming map measurements into real-life distances. This means you can rely on the map to accurately reflect the space between two locations without physically measuring it outside.

Map scales often use inches or centimeters as units, while actual distances might be represented in miles or kilometers. Therefore, understanding the map's scale is essential for accurate distance measurement. It allows you to have a correct understanding of the layout of a geographic area without distortion, relying solely on the given proportionality.

Algebraic Equations

Algebraic equations come in handy when you're trying to solve problems involving proportionality, such as determining the constant of proportionality. By setting up an equation, we can relate two quantities – in this case, the distance on a map and the actual distance in the real world – using algebra.

Here, we start by letting the map distance be represented as "M" and the actual distance as "d". The given problem shows us that these two distances are related by a constant ratio or 'k'. This relationship is represented by the equation \( M = k \times d \). In our example, the equation becomes \( \frac{1}{2} = k \times 5 \).

Solving equations often involves isolating 'k' (or the unknown) by performing operations such as division or multiplication. In this instance, dividing both sides by 5 helps uncover the constant of proportionality \( k = \frac{1}{10} \). Doing this correctly ensures that we can find consistent and useful results that can be applied elsewhere, demonstrating the power of algebraic thinking.