Question:** A soil scientist studying erosion patterns near a river observes a triangular plot of land with side lengths of 13 m, 14 m, and 15 m. To analyze drainage flow, she needs the length of the shortest altitude. What is the length of the shortest altitude?

["Question: A soil scientist studying erosion patterns near a river observes a triangular plot of land with side lengths of 13 m, 14 m, and 15 m. To analyze drainage flow, she needs the length of the shortest altitude. What is the length of the shortest altitude?", "Understanding the Problem\nFor effective soil and water management, understanding a triangular plot’s geometry is essential. The shortest altitude corresponds to the longest side, as altitude decreases with increased base length for a fixed area. Here, the triangle has sides 13 m, 14 m, and 15 m—making it a nearly right-angled triangle—so the longest side (15 m) is most likely the base for the shortest altitude.", "In this article, we’ll calculate the area using Heron’s formula and then determine the altitude to the longest side to find the shortest altitude.", "---", "### Step 1: Compute the Semi-Perimeter\nGiven side lengths:\n( a = 13 , \ ext{m}, , b = 14 , \ ext{m}, , c = 15 , \ ext{m} )", "The semi-perimeter ( s ) is:\n[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 , \ ext{m}\n]", "---", "### Step 2: Calculate the Area Using Heron’s Formula\nHeron’s formula states:\n[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substitute values:\n[\n\ ext{Area} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculate:\n[\n21 \ imes 8 = 168,\quad 7 \ imes 6 = 42,\quad 168 \ imes 42 = 7056\n]\n[\n\ ext{Area} = \sqrt{7056} = 84 , \ ext{m}^2\n]", "---", "### Step 3: Find the Altitude to the Longest Side (15 m)\nArea of a triangle is also given by:\n[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Using base = 15 m:\n[\n84 = \frac{1}{2} \ imes 15 \ imes h \quad \Rightarrow \quad 84 = 7.5h\n]\n[\nh = \frac{84}{7.5} = 11.2 , \ ext{m}\n]", "---", "### Step 4: Verify Altitudes to Other Sides\nSince the shortest altitude corresponds to the longest base:\n- Altitude to side 14 m:\n[\nh = \frac{2 \ imes 84}{14} = \frac{168}{14} = 12 , \ ext{m}\n]\n- Altitude to side 13 m:\n[\nh = \frac{168}{13} \approx 12.92 , \ ext{m}\n]", "These confirm that the altitude to the 15 m side is indeed the smallest.", "---", "### Why This Matters for the Soil Scientist\nUnderstanding the shortest drainage altitude helps predict water flow and erosion risk. A shorter altitude implies steeper drainage, potentially increasing erosion on the 15 m side during heavy rainfall. Mapping these altitudes supports targeted conservation strategies such as contour bunding, vegetation strips, or check dams.", "---", "Conclusion\nThe shortest altitude in the 13–14–15 m triangular plot is 11.2 meters, corresponding to the altitude drawn to the 15 m side. Accurate measurements like this empower soil scientists to model water movement, optimize land use, and prevent soil degradation in riverine environments.", "Key Terms:\n- Triangle area calculation\n- Heron’s formula\n- Shortest altitude in a triangle\n- Soil erosion analysis\n- Riverine land management"]