\boxed{19}Question: A meteorologist models the change in atmospheric pressure over a 24-hour period with the function $ P(t) = t^3 - 6t^2 + 9t + 10 $, where $ t $ represents time in hours after midnight. At what time(s) does the pressure reach a local minimum or maximum?
["Understanding Local Extrema: When Does Atmospheric Pressure Peaks or Lows?", "Weather patterns are complex, but understanding atmospheric pressure changes helps meteorologists predict storms, winds, and climate behavior. In this example, the pressure variation over a day is modeled by the cubic function:", "[\nP(t) = t^3 - 6t^2 + 9t + 10\n]", "where $ t $ is time in hours after midnight. To find when the pressure reaches a local minimum or local maximum, we use calculus—specifically, by analyzing the function’s derivative.", "---", "### Step 1: Find the First Derivative", "To locate critical points where extrema may occur, take the derivative of $ P(t) $:", "[\nP'(t) = \frac{d}{dt}(t^3 - 6t^2 + 9t + 10) = 3t^2 - 12t + 9\n]", "---", "### Step 2: Solve for Critical Points", "Set $ P'(t) = 0 $ to find critical points:", "[\n3t^2 - 12t + 9 = 0\n]", "Divide through by 3:", "[\nt^2 - 4t + 3 = 0\n]", "Factor the quadratic:", "[\n(t - 1)(t - 3) = 0\n]", "So, $ t = 1 $ and $ t = 3 $ are the critical points.", "---", "### Step 3: Use the Second Derivative Test to Classify Extrema", "Find the second derivative:", "[\nP''(t) = \frac{d}{dt}(3t^2 - 12t + 9) = 6t - 12\n]", "Evaluate $ P''(t) $ at each critical point:", "- At $ t = 1 $:\n [\n P''(1) = 6(1) - 12 = -6 < 0 \Rightarrow \ ext{Local maximum}\n ]", "- At $ t = 3 $:\n [\n P''(3) = 6(3) - 12 = 6 > 0 \Rightarrow \ ext{Local minimum}\n ]", "---", "### What Do These Times Mean Weather-wise?", "- At $ t = 1 $ hour after midnight (1:00 AM), atmospheric pressure reaches a local maximum, suggesting a brief but significant pressure rise, possibly preceding high wind or storm activity.\n- At $ t = 3 $ hours after midnight (3:00 AM), pressure hits a local minimum, indicating a drop in atmospheric pressure—often a precursor to precipitation or low-pressure systems forming.", "---", "### Why This Matters in Meteorology", "Identifying local extrema helps forecasters anticipate rapid weather changes. While cubic models are simplified, they capture essential trends like pressure spikes and dips. Combining such analysis with real-time data improves forecast accuracy.", "---", "In summary, using derivatives, we find the 24-hour pressure function $ P(t) = t^3 - 6t^2 + 9t + 10 $ has a local maximum at t = 1 and a local minimum at t = 3, offering key insight into atmospheric behavior. Understanding these moments empowers weather prediction and public safety planning."]
