If The Actual Distance Between The Plaza De Armas Of Cusco And Machu Picchu Is 64.7 Km, How Many Centimeters Will It Measure On A Map With A Scale Of 1:500,000?

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Introduction

In this article, we'll explore how to calculate the distance on a map between two locations, specifically the Plaza de Armas in Cusco and Machu Picchu, given the actual distance and the map's scale. This is a practical application of understanding scale ratios and unit conversions, essential skills in geography, cartography, and even everyday problem-solving. We will tackle the problem step-by-step, ensuring clarity and accuracy in our calculations. The problem we're addressing is: If the actual geometric distance between the Plaza de Armas in Cusco and Machu Picchu is 64.7 km, how many centimeters will this distance measure on a map with a scale of 1:500,000? This involves converting kilometers to centimeters and applying the scale ratio to find the corresponding map distance. Understanding map scales is fundamental in interpreting distances on maps accurately. A map scale represents the ratio between a distance on a map and the corresponding distance on the ground. For instance, a scale of 1:500,000 means that 1 unit of measurement on the map represents 500,000 units of the same measurement on the Earth's surface. This concept is crucial in various fields, including navigation, urban planning, and environmental studies. Accurate map reading and distance calculation are essential for effective decision-making in these areas. Moreover, understanding scales helps us appreciate the level of detail a map can provide and the trade-offs between scale and the amount of information a map can display. Maps with larger scales (e.g., 1:25,000) show smaller areas with greater detail, while maps with smaller scales (e.g., 1:1,000,000) cover larger areas but with less detail. The ability to convert between real-world distances and map distances is a valuable skill in various contexts. In this article, we will break down the process into manageable steps, making it easy to follow and understand. By the end of this guide, you will be equipped to solve similar problems involving map scales and distance calculations.

Understanding the Problem

Before diving into the solution, it's crucial to clearly understand the problem we're trying to solve. We are given the actual distance between the Plaza de Armas in Cusco and Machu Picchu, which is 64.7 kilometers. We also know that Gustavo's map has a scale of 1:500,000. This scale tells us the ratio between the distance on the map and the corresponding distance in reality. Our goal is to determine how many centimeters this 64.7 km distance will measure on the map. To do this, we need to convert the real-world distance into the same units as the map scale (centimeters) and then apply the scale ratio. This involves a few key steps: first, converting kilometers to meters, then meters to centimeters, and finally, using the scale to find the map distance. Misunderstanding the problem or skipping a conversion step can lead to significant errors in the final result. Therefore, it's essential to approach the problem systematically, breaking it down into smaller, more manageable parts. Careful reading and comprehension are key to success in solving mathematical problems, especially those involving real-world applications like map scales. Understanding the relationship between map distances and real-world distances is crucial for various applications, including navigation, planning, and geographical analysis. The scale of a map dictates how accurately distances can be represented, and choosing the right map scale depends on the purpose of the map. For example, a tourist map might use a smaller scale to show a larger area, while a hiking map would use a larger scale to provide more detail. In this particular problem, the scale of 1:500,000 indicates that the map will cover a relatively large area, but it still provides a reasonable level of detail for understanding the distance between two significant locations like Cusco and Machu Picchu. By solving this problem, we not only find the map distance but also reinforce our understanding of how map scales work and how they are used to represent the world around us. The ability to work with scales and conversions is a fundamental skill in both mathematics and geography, and it has practical applications in many real-life scenarios.

Step 1: Convert Kilometers to Centimeters

The first step in solving this problem is to convert the actual distance from kilometers to centimeters. This is necessary because the map scale is given in a ratio that implies the same units on both sides (in this case, centimeters). We know that 1 kilometer is equal to 1000 meters, and 1 meter is equal to 100 centimeters. Therefore, to convert kilometers to centimeters, we need to multiply by 1000 to convert to meters and then multiply by 100 to convert to centimeters. This can be done in a single step by multiplying by 100,000 (1000 * 100). So, 64.7 kilometers is equal to 64.7 * 100,000 centimeters. Performing this multiplication gives us 6,470,000 centimeters. This is the actual distance between the Plaza de Armas in Cusco and Machu Picchu expressed in centimeters. Accurate unit conversions are crucial in any scientific or mathematical calculation, and this step ensures that we are working with consistent units. Incorrect unit conversions are a common source of errors, so it's important to double-check your calculations and use the correct conversion factors. In this case, understanding the relationship between kilometers, meters, and centimeters is essential. The conversion factor of 100,000 cm per kilometer is a fundamental piece of information needed to solve this problem. By converting the distance to centimeters, we can now directly compare it to the scale of the map, which is also expressed in terms of centimeters. This allows us to use the scale ratio to find the corresponding distance on the map. The process of converting units is a fundamental skill in many areas of science and engineering. It allows us to express measurements in different scales and to compare quantities that are initially given in different units. In the context of map reading and cartography, accurate unit conversions are essential for determining real-world distances from map measurements and vice versa. By mastering this skill, we can confidently work with maps and scales to solve a variety of practical problems.

Step 2: Apply the Map Scale

Now that we have the actual distance in centimeters (6,470,000 cm), we can apply the map scale to find the corresponding distance on the map. The map scale is given as 1:500,000, which means that 1 centimeter on the map represents 500,000 centimeters in reality. To find the distance on the map, we need to divide the actual distance by the scale factor. In this case, we divide 6,470,000 cm by 500,000. This calculation will give us the distance on the map in centimeters. Performing the division, 6,470,000 / 500,000, results in 12.94 centimeters. This means that the distance between the Plaza de Armas in Cusco and Machu Picchu will measure 12.94 centimeters on Gustavo's map. Understanding how to apply map scales is crucial for accurately interpreting distances represented on maps. The scale factor acts as a divisor, reducing the real-world distance to its corresponding map distance. This is a fundamental concept in cartography and map reading. Accurate application of the scale ensures that the map remains a reliable representation of the physical world. Map scales can be expressed in different ways, including as a ratio (like 1:500,000), as a fraction (1/500,000), or as a verbal statement (e.g.,