5**Question:** A meteorologist is modeling the temperature changes over a day and defines a function \( T(t) = 3t^2 - 12t + 20 \), where \( T \) is the temperature in degrees Celsius and \( t \) is the time in hours from midnight. Determine the time \( t \) when the temperature reaches its minimum.

["Modeling Daily Temperature: When Does Your Temperature Peak? A Deep Dive Using Quadratic Functions", "Understanding how temperature varies throughout the day is essential in meteorology, climate science, and everyday planning. In this article, we explore a real-world application of quadratic functions by analyzing a temperature model defined as:", "[\nT(t) = 3t^2 - 12t + 20\n]", "where ( T ) is the temperature in degrees Celsius and ( t ) is the time in hours after midnight. This equation describes a continuous temperature trend, ideal for identifying key moments like minimum or maximum temperatures.", "One of the most common questions meteorologists face is: When does the temperature reach its minimum during the day? Since this function is quadratic and opens upward (the coefficient of ( t^2 ) is positive), it has a single minimum point, which we can precisely determine using calculus or algebra. Let’s walk through how to find that critical time.", "### Understanding the Quadratic Model", "The general form of a quadratic function is:", "[\nT(t) = at^2 + bt + c\n]", "In our case, ( a = 3 ), ( b = -12 ), and ( c = 20 ). Because ( a > 0 ), the parabola opens upward, confirming the existence of a minimum point. The vertex of this parabola gives the exact time when temperature is lowest.", "### Finding the Vertex: When Does Minimum Temperature Occur?", "The formula for the time ( t ) at the vertex (minimum point) is:", "[\nt = -\frac{b}{2a}\n]", "Substituting the values:", "[\nt = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2\n]", "Thus, the minimum temperature occurs at ( t = 2 ) hours after midnight — exactly 2:00 AM.", "### Confirming with First Derivative (Optional Math Depth)", "To reinforce understanding, we can differentiate ( T(t) ):", "[\nT'(t) = 6t - 12\n]", "Setting the derivative to zero to find critical points:", "[\n6t - 12 = 0 \Rightarrow t = 2\n]", "Since the second derivative ( T''(t) = 6 > 0 ), this critical point is indeed a minimum.", "### Practical Implications", "At ( t = 2 ) hours, or 2:00 AM, the temperature reaches its monthly minimum of:", "[\nT(2) = 3(2)^2 - 12(2) + 20 = 12 - 24 + 20 = 8^\circ\ ext{C}\n]", "While this might seem cool, real-world models often refine such curves further, but mathematically, this is where the dip occurs.", "### Final Takeaway", "Modeling temperature with quadratic functions like ( T(t) = 3t^2 - 12t + 20 ) allows meteorologists to predict critical changes in weather patterns efficiently. Using the vertex formula, we find that minimum temperature occurs at 2 hours after midnight — a key insight for weather forecasting, agricultural planning, and daily lifestyle adjustments.", "---", "Key Takeaway: By analyzing the coefficient of ( t^2 ) and applying ( t = -\frac{b}{2a} ), you can quickly determine the exact time when a quadratic model predicts a minimum temperature. This mathematical approach empowers accurate weather modeling."]