The height is $ 3k = 3\sqrt{34} $. Numerically, $ \sqrt{34} pprox 5.83 $, so height $ pprox 17.49 $, but exact form is required: $oxed{3\sqrt{34}}$ inches.
["The Exact Height: Understanding Why It’s Written as $ \boxed{3\sqrt{34}} $ Inches", "When solving for height given the equation $ 3k = 3\sqrt{34} $, many may initially simplify the expression and fall into the common trap of approximating $ \sqrt{34} \approx 5.83 $, yielding a numerical height of about 17.49 inches. However, in mathematics—especially when precision matters—using the exact form is essential.", "The equation $ 3k = 3\sqrt{34} $ simplifies directly to $ k = \sqrt{34} $. Therefore, the exact height is:", "$$\n\boxed{3\sqrt{34}} \ ext{ inches}\n$$", "This symbolic representation preserves mathematical integrity and allows for exact calculations, trigonometric applications, or further algebraic manipulation without loss of accuracy.", "### Why approximate values fall short\nWhile $ \sqrt{34} \approx 5.83 $ gives a quick estimate, exact expressions like $ 3\sqrt{34} $ are critical in fields such as geometry, engineering, and physics, where precision impacts design and analysis. Using the radical form prevents propagation of rounding errors and supports symbolic computation.", "### Beyond approximation: Practical exact use\nThis exact form appears naturally in scenarios involving:\n- Height-to-width ratios\n- Diagonal measurements in rectangular prisms\n- Vector projections requiring precise scaling", "For example, if designing a display case with a diagonal height modeled by $ 3\sqrt{34} $ inches, engineers rely on the precise value to ensure structural compatibility and material uniformity.", "### Conclusion\nWhile numerical approximations offer convenience, $ \boxed{3\sqrt{34}} $ inches is the mathematically precise and recommended form for this height. Embracing exact values maintains clarity and rigor, demonstrating the importance of exact expressions in professional and scientific contexts.", "Use precise notation—because accuracy matters."]
