Problem 12 - Technical Mathematics with Calculus

Question

A U.S. Geological Survey map has a scale of \(1: 24,000 .\) If two buildings are 3.86 miles apart, how many inches apart are they on the map?

Step-by-Step Solution

Answer

The buildings are 10.19 inches apart on the map. (3.86 miles * 63,360 inches/mile) / 24,000 = 10.19 inches.

Step 1: Understand the Scale

The scale on a map indicates the ratio of a distance on the map to the corresponding distance on the ground. A scale of 1:24,000 means that 1 unit on the map is equal to 24,000 units in reality.

Step 2: Convert Miles to Inches

Before applying the scale, convert the distance between the buildings from miles to inches, knowing that 1 mile equals 63,360 inches (1 mile = 5,280 feet and 1 foot = 12 inches, hence 5,280 feet/mile * 12 inches/foot).

Step 3: Calculate the Actual Distance in Inches

To find the actual distance in inches between the buildings, multiply the distance in miles by the number of inches in a mile: 3.86 miles * 63,360 inches/mile.

Step 4: Apply the Scale to Find the Map Distance

Now, use the scale to find the distance on the map. Divide the actual distance in inches by the scale factor: (Actual distance in inches) / 24,000.

Key Concepts

Converting Miles to Inches

When working with maps and measurements, one may need to convert large-scale distances into a more manageable unit such as inches. To do this, it's crucial to know the standard conversion rates. There are 5,280 feet in a mile, and since there are 12 inches in a foot, a mile can be converted to inches by multiplying it with both these numbers.

For instance, to convert 3.86 miles to inches, the process would involve the following calculation: \[ 3.86 \text{ miles} \times 5,280 \text{ feet/mile} \times 12 \text{ inches/foot} = 3.86 \times 63,360 \text{ inches} \].This conversion is the first step in utilizing the map scale to calculate the actual distance on the map between two points.

Understanding Map Scales

A map scale is essential for understanding the proportional relationship between a distance on a map and the actual distance on the ground. It is represented as a ratio such as '1:24,000’, which means that one unit of measurement on the map (be it inches or centimeters) is equal to 24,000 of the same units in reality.

When attempting to determine the map distance between two points, applying the scale factor is necessary. If 1 inch on the map represents 24,000 inches in the real world, you can find out the map distance by dividing the real-world distance in inches by the scale factor. In this case, the calculation for a distance that covers 3.86 miles in reality (which was already converted to inches) will look as follows: \[ (\text{Actual distance in inches}) \div 24,000 \].This result gives you the linear measurement between the two points on the map.

Geographic Distance Representation

Representing geographic distance on maps is a way to visualize large-scale distances in a much smaller, and more convenient format. Maps are a scaled-down depiction of the world or a section of it and understanding how to interpret them is important for navigation, planning, and comprehending spatial relationships.

To translate real-world measurements onto a map, converting them into a scale that the map uses is indispensable. When you're given the scale, such as '1:24,000’, it allows for precise calculation of how a distance, like the one between two buildings, is represented on this paper, or digital medium. This abstract conversion bridges the gap between the physical world and its two-dimensional representation, making it possible to grasp complex geographic data simply by looking at a drawing or printout.