#### 2, \(-\frac{1}{2}\)Question: A weather radar detects a storm front forming a triangle with sides measuring 10 km, 13 km, and 15 km. What is the length of the shortest altitude?

["The Shortest Altitude of a Triangle Formed by Storm Fronts: Solving a Real-World Geometry Problem", "When a weather radar detects a developing storm front, meteorologists and geoscientists often analyze its spatial boundaries—sometimes visualized as a triangular region. In one such scenario, a storm front forms a triangle with sides measuring 10 km, 13 km, and 15 km. Understanding the triangle’s geometry helps estimate triangular storm boundaries and calculate essential metrics, including altitudes, which can correlate with storm intensity and movement.", "In this article, we’ll explore how to compute the shortest altitude of a triangle with side lengths 10 km, 13 km, and 15 km. This calculation is crucial not only in math education but also in applied fields like weather prediction, where precise area and height estimations inform storm tracking and risk assessment.", "---", "### Step 1: Verify the Triangle Is Valid", "Before calculating altitudes, confirm the sides form a valid triangle using the triangle inequality:", "- (10 + 13 > 15) → (23 > 15) ✔️\n- (10 + 15 > 13) → (25 > 13) ✔️\n- (13 + 15 > 10) → (28 > 10) ✔️", "All conditions are satisfied, so a real triangle exists.", "---", "### Step 2: Compute the Area Using Heron’s Formula", "The shortest altitude corresponds to the longest side, since altitude ( h = \frac{2A}{\ ext{base}} ). To minimize altitude, we use the longest side (15 km) as the base.", "Heron’s formula calculates the area ( A ) from side lengths ( a = 10 ), ( b = 13 ), ( c = 15 ):", "[\ns = \frac{a + b + c}{2} = \frac{10 + 13 + 15}{2} = \frac{38}{2} = 19 \ ext{ km}\n]", "[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{19(19 - 10)(19 - 13)(19 - 15)} = \sqrt{19 \ imes 9 \ imes 6 \ imes 4}\n]", "[\nA = \sqrt{19 \ imes 9 \ imes 24} = \sqrt{4104} \approx 64.07 \ ext{ km}^2\n]", "---", "### Step 3: Calculate the Shortest Altitude", "Using the base ( c = 15 ) km:", "[\nh_{\ ext{min}} = \frac{2A}{c} = \frac{2 \ imes \sqrt{4104}}{15} \approx \frac{2 \ imes 64.07}{15} = \frac{128.14}{15} \approx 8.54 \ ext{ km}\n]", "Thus, the shortest altitude is approximately 8.54 km, corresponding to the side opposite the largest angle—typically the storm front’s most active boundary.", "---", "### Why This Matters in Weather Forecasting", "Knowing the storm’s effective height (altitude) aids meteorologists in modeling storm dynamics. Larger altitudes may indicate higher cloud formations, stronger updrafts, and more intense weather phenomena. By computing altitudes from actual dimensions—via tools like Heron’s formula—forecasters enhance accuracy in risk prediction and public safety messaging.", "---", "### Final Answer", "The shortest altitude of the storm front triangle with sides 10 km, 13 km, and 15 km is approximately 8.54 km, corresponding to the altitude relative to the 15 km side.", "---", "Keywords: storm triangle altitude, weather radar geometry, Heron’s formula, shortest altitude calculation, triangle storm boundaries, atmospheric modeling, triangle height storm front, meteorology math, geometric storm analysis.", "---", "Summary\nBy applying Heron’s formula to compute area from side lengths, and identifying the shortest altitude relative to the longest side, we precisely determine critical storm characteristics. This fusion of geometry and meteorology enhances storm understanding and aids real-time forecasting."]