Solution: The cosine function oscillates between $ -1 $ and $ 1 $, so $ 5\cos\left(\frac{\pi}{6}x\right) $ ranges from $ -5 $ to $ 5 $. Adding 10 shifts the range to $ [5, 15] $. The range is $ \boxed{[5, 15]} $.

["Understanding How Shifting the Cosine Function Expands Its Range: From $[-1, 1]$ to $[5, 15]$", "The cosine function is a fundamental concept in trigonometry, known for its predictable oscillation between $-1$ and $1$. Mathematically, for any real number $x$,\n$$\n\cos(x) \in [-1, 1]\n$$\nThis well-defined range becomes the foundation for understanding transformed versions of the cosine function, such as $ 5\cos\left(\frac{\pi}{6}x\right) $, and how shifts affect its output.", "When we multiply the cosine function by $5$, the amplitude increases, stretching the output intervals:\n$$\n5\cos\left(\frac{\pi}{6}x\right) \in [-5, 5]\n$$\nThis scaling compresses the wave vertically but preserves the $5$ units width around zero.", "But what happens when we shift this entire transformed cosine function upward? Adding $10$ translates the function’s vertical position:\n$$\n5\cos\left(\frac{\pi}{6}x\right) + 10 \in [-5 + 10, 5 + 10] = [5, 15]\n$$\nThis vertical shift effectively redefines the function’s range, moving it from centered oscillation to a scaled and raised interval spanning $[5, 15]$.", "So, the complete expression $ 5\cos\left(\frac{\pi}{6}x\right) + 10 $ maps the oscillating cosine wave into a fixed interval where its minimum value is $5$ and maximum reaches $15$. This refined range is mathematically represented as:\n$$\n\boxed{[5, 15]}\n$$", "Understanding how amplitude scaling and vertical shifting influence periodic functions helps students and professionals alike interpret transformations in trigonometry with clarity and precision. Whether modeling waves, oscillations, or cyclic phenomena, recognizing these ranges ensures accurate analysis and predictive modeling across physics, engineering, and data science."]