A cartographer is calculating the gradient of a mountain trail that rises from 850 meters to 2,150 meters over a horizontal distance of 12.5 kilometers. What is the percent grade of the trail?
["Calculating the Percent Grade of a Mountain Trail: A Cartographer’s Guide", "When mapping rugged terrain, one essential calculation is determining the gradient or steepness of a trail — especially vital for hikers, trail planners, and researchers. For example, a cartographer recently analyzed a mountainous path that ascends from 850 meters to 2,150 meters over a horizontal distance of 12.5 kilometers. Understanding the slope’s percent grade helps assess trail difficulty and design safer routes.", "### Understanding Gradient and Percent Grade", "The gradient of a trail refers to its steepness, expressed as a ratio of vertical rise to horizontal distance. The percent grade converts this ratio into a percentage, making it intuitive for interpreting slope intensity. It’s calculated using the formula:", "[\n\ ext{Percent Grade} = \left( \frac{\ ext{Vertical Rise}}{\ ext{Horizontal Distance (in same units)}} \right) \ imes 100%\n]", "### Step-by-Step Calculation", "Let’s apply this to the mountain trail:", "- Vertical Rise (m):\n ( 2,150,\ ext{m} - 850,\ ext{m} = 1,300,\ ext{m} )", "- Horizontal Distance:\n Given as 12.5 kilometers. Convert this to meters for consistency:\n ( 12.5,\ ext{km} = 12,500,\ ext{m} )", "- Apply the percent grade formula:\n [\n \ ext{Percent Grade} = \left( \frac{1,300}{12,500} \right) \ imes 100%\n ]", "- Perform the division:\n ( \frac{1,300}{12,500} = 0.104 )", "- Convert to percentage:\n ( 0.104 \ imes 100% = 10.4% )", "### Result Interpretation", "The trail has a percent grade of 10.4%, indicating a steep incline. On a map or navigation guide, this means every 100 meters traveled horizontally, the trail rises 10.4 meters vertically. For cartographers, this precise calculation enhances trail descriptions, supports accurate contour mapping, and aids in safety planning and route planning for outdoor enthusiasts.", "### Final Notes", "Understanding gradient percentages enables better trail design and user safety—critical in mountainous terrain where steep slopes pose significant challenges. Whether used in GIS mapping, topographic surveys, or trail construction, this calculation remains a fundamental tool in cartography and outdoor recreation.", "---", "Keywords: percent grade mountain trail, gradient calculation elevation rise, cartography slope percentage, hiking trail steepness, vertical rise horizontal distance, mapping technical data, percent grade formula."]
