A climate change journalist observes a triangular seawall with side lengths 13 m, 14 m, and 15 m. To assess its structural integrity, she needs to determine the length of the shortest altitude. What is the length of the shortest altitude, in meters?

["The Shortest Altitude of a Triangular Seawall: A Climate-Resilient Structure Analyzed", "When designing or evaluating coastal defenses like seawalls, understanding their structural integrity is critical—especially in an era shaped by climate change and rising sea levels. A recent observation by a climate change journalist examining a triangular seawall with side lengths 13 m, 14 m, and 15 m highlights an essential engineering calculation: determining the shortest altitude.", "The shortest altitude of any triangle corresponds to the longest side, since the altitude invertively relates to area: the larger the base, the smaller the required height to maintain the same area. For this seawall, the side lengths 13 m, 14 m, and 15 m form a well-known scalene triangle, often used in geometry problems due to its integer values and Pythagorean-triangle properties.", "To find the shortest altitude, we first compute the area using Heron’s formula, then use the area and the longest side to determine the shortest altitude.", "---", "### Step 1: Compute the semi-perimeter", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \ ext{ m}\n]", "---", "### Step 2: Compute the area using Heron’s formula", "[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n]\n[\n= \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]\n[\n= \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84 \ ext{ m}^2\n]", "---", "### Step 3: Use the area to find altitudes", "Area is also given by:\n[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n\Rightarrow h = \frac{2 \ imes \ ext{Area}}{\ ext{base}}\n]", "Compute altitudes relative to each side:", "- Altitude to side 13 m:\n[\nh_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92 \ ext{ m}\n]", "- Altitude to side 14 m:\n[\nh_{14} = \frac{2 \ imes 84}{14} = \frac{168}{14} = 12 \ ext{ m}\n]", "- Altitude to side 15 m (longest side, shortest height expected):\n[\nh_{15} = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2 \ ext{ m}\n]", "---", "### Conclusion: The shortest altitude is 11.2 meters", "This result confirms that the shortest altitude, relative to the longest base of 15 m, is ( \frac{168}{15} = \frac{56}{5} = 11.2 ) m. From a climate resilience perspective, monitoring such structural features ensures seawalls remain effective against storm surges and sea-level rise.", "For coastal communities and engineers, triangular seawalls like this—engineered with precise geometry—stand as quiet sentinels against a changing climate. Understanding their internal stresses through elegant math ensures safer, longer-lasting infrastructure.", "Short answer:\nThe length of the shortest altitude is 11.2 meters."]