Solution: The maximum value of $ \sin $ is 1, so the maximum depth occurs when $ \sin\left(\frac{\pi t}{12}\right) = 1 $. Substituting, $ d_{\text{max}} = 10 + 4(1) = 14 $. \boxed{14}
["Understanding Maximum Depth Using the Sine Function: How $ \sin\left(\frac{\pi t}{12}\right) = 1 $ Determines Optimal Value", "In mathematical modeling and signal analysis, identifying the maximum value of periodic functions is essential for optimizing systems, particularly when dealing with time-varying processes. One key example is analyzing wave-like phenomena governed by the sine function. This article explores how the maximum value of $ \sin x $—which is 1—drives the calculation of maximum depth in a sinusoidal depth model.", "### How the Sine Function Defines Maximum Depth", "The sine function, $ \sin(x) $, oscillates between $-1$ and $1$ across all real numbers. In models representing depth influenced by periodic behavior—such as depth fluctuations in environmental or engineering systems—the maximum attainable depth corresponds directly to the peak value of $ \sin x $, which is exactly 1.", "This maximum occurs when the argument satisfies:", "[\n\sin\left(\frac{\pi t}{12}\right) = 1\n]", "Solving for $ t $:", "[\n\frac{\pi t}{12} = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}\n]", "The smallest positive $ t $ occurs when $ k = 0 $, giving:", "[\n\frac{\pi t}{12} = \frac{\pi}{2} \quad \Rightarrow \quad t = 6\n]", "This means maximum depth physical relevance happens at $ t = 6 $ hours in the model’s cycle.", "### Computing Maximum Depth $ d_{\ ext{max}} $", "Using the model equation that incorporates the sine term, such as:", "[\nd(t) = 10 + 4 \sin\left(\frac{\pi t}{12}\right)\n]", "At $ t = 6 $, where $ \sin\left(\frac{\pi \cdot 6}{12}\right) = \sin\left(\frac{\pi}{2}\right) = 1 $, the equation reaches its peak:", "[\nd_{\ ext{max}} = 10 + 4(1) = 14\n]", "This confirms that the maximum depth is 14 units—whether modeling ocean tides, sensor depth shifts, or similar periodic dynamics.", "### Conclusion", "By leveraging the fundamental property that $ \max(\sin x) = 1 $, we precisely determine when depth reaches its maximum value in sinusoidal models. Substituting this at the optimal $ t $ yields the maximum depth:", "[\n\boxed{14}\n]", "Understanding this relationship enhances modeling accuracy and supports effective decision-making in fields ranging from environmental science to mechanical engineering. When dealing with periodic depth variations, always check where the sine function peaks to capture the true maximum."]
