Question: An aerospace engineer is analyzing the trajectory of a rocket modeled as a triangle with side lengths $ 13 $, $ 14 $, and $ 15 $ kilometers. What is the length of the longest altitude of this triangle?
["1. An Aerospace Engineer’s Triangle: More Than a Shape — A Key to Rocket Precision \nIn the evolving landscape of aerospace design, even a triangle — simple in form — carries deep significance. Recently, engineers modeling rocket trajectories have turned to a 13-14-15 kilometer triangular framework not just for structural insights, but to optimize altitude and thrust efficiency. When tasked with calculating the longest altitude of this precise triangular cross-section, curiosity grows: how does geometry inform real-world flight performance? This question now surfaces at the intersection of mathematics, aerodynamics, and innovation — and matters more in the US aerospace sector than ever.", "Why the 13-14-15 Triangle Matters in Rocket Trajectory Design \nThe 13-14-15 triangle is beloved in geometric study due to its well-defined properties. With sides measuring 13, 14, and 15 kilometers — a scalable model resembling certain launch configurations — engineers analyze this shape to understand force distribution, lift dynamics, and flight stability. Within the context of rocket aerodynamics, calculating altitudes derived from triangle altitudes influences decision-making around fuel loading, casing stress, and atmospheric entry angles. This triangle becomes a foundational tool, enabling precise trajectory modeling in an industry driven by data and simulation.", "How to Calculate the Longest Altitude: A Clear, Step-by-Step Breakdown \nTo determine the longest altitude, we first compute the triangle’s area — a critical step for altitude calculations. Using Heron’s formula, the semi-perimeter $ s = \frac{13 + 14 + 15}{2} = 21 $. The area is then: \n$$\nA = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21 \cdot (21-15) \cdot (21-14) \cdot (21-13)} = \sqrt{21 \cdot 6 \cdot 7 \cdot 8} = \sqrt{7056} = 84 \ ext{ km}^2.\n$$ \nEach altitude corresponds to $ h = \frac{2A}{\ ext{base}} $. The longest altitude occurs on the shortest side — here, the 13-kilometer base: \n$$\nh_{\ ext{max}} = \frac{2 \ imes 84}{13} = "]
