#### 95**Question:** An ornithologist is tracking bird migration patterns and models the position of a bird using the function \( h(t) = 4t^2 - 3t + 2 \). Find the time \( t \) when the bird is at its minimum height during the flight from \( t = 0 \) to \( t = 3 \).

["How Ornithologists Model Bird Migration: Finding the Minimum Height Using Quadratic Functions", "When studying bird migration, ornithologists often rely on mathematical models to understand key behaviors such as flight altitude over time. One classic example is tracking a bird’s vertical position using a quadratic function. Consider the position of a bird modeled by:", "[\nh(t) = 4t^2 - 3t + 2\n]", "This function describes height ( h(t) ) in meters as a function of time ( t ) in hours, from ( t = 0 ) to ( t = 3 ). Understanding when the bird reaches its minimum height during this period is crucial for analyzing energy use and flight efficiency.", "### Why Quadratic Models Work for Bird Altitude", "Bird flight patterns—especially during flapping or gliding phases—often follow smooth, curved paths. A parabolic function like ( h(t) = 4t^2 - 3t + 2 ) is ideal for modeling vertical displacement over time, capturing gradual changes in elevation without abrupt jumps. The coefficient of ( t^2 ) is positive (4), meaning the trajectory opens upward, indicating the bird climbs gradually from lower altitudes before eventually rising to a peak.", "### Finding the Minimum Height Using Calculus", "To find the time when the bird is at minimum height within the interval ( [0, 3] ), we analyze the function using calculus. Since ( h(t) ) is a continuous quadratic, its minimum on a closed interval occurs either at a critical point inside the interval or at an endpoint.", "Step 1: Compute the derivative\nThe first derivative ( h'(t) ) gives the rate of change of height:", "[\nh'(t) = \frac{d}{dt}(4t^2 - 3t + 2) = 8t - 3\n]", "Step 2: Find critical points\nSet ( h'(t) = 0 ) to locate possible minima:", "[\n8t - 3 = 0 \implies t = \frac{3}{8} = 0.375 \ ext{ hours}\n]", "This critical point lies within the interval ( [0, 3] ), so it must be evaluated.", "Step 3: Evaluate height at critical point and endpoints", "- At ( t = 0 ):\n [\n h(0) = 4(0)^2 - 3(0) + 2 = 2 \ ext{ meters}\n ]", "- At ( t = 0.375 ):\n [\n h(0.375) = 4(0.375)^2 - 3(0.375) + 2 = 4(0.140625) - 1.125 + 2 = 0.5625 - 1.125 + 2 = 1.4375 \ ext{ meters}\n ]", "- At ( t = 3 ):\n [\n h(3) = 4(3)^2 - 3(3) + 2 = 4(9) - 9 + 2 = 36 - 9 + 2 = 29 \ ext{ meters}\n ]", "Conclusion:\nThe bird reaches its minimum height of ( 1.4375 ) meters at ( t = 0.375 ) hours (approximately 22.5 minutes) during the first 3 hours of flight. This minimum occurs because ( h(t) ) has a minimum at ( t = \frac{3}{8} ), confirmed by the positive curvature sign and endpoint comparisons.", "### Practical Implication for Ornithologists", "Knowing the exact time of minimum altitude helps researchers correlate bird behavior with environmental factors—such as avoiding wind gusts at lower altitudes or optimizing energy use during ascent. The quadratic model not only predicts trajectory but also informs conservation strategies by revealing key waypoints in a bird’s migratory journey.", "---", "Key Takeaways:\n- Model bird height with a quadratic function for smooth trajectory analysis.\n- Use derivatives to locate critical points and find minima.\n- Always evaluate endpoints to confirm global minima within a time interval.\n- Mathematical modeling enhances understanding of avian flight mechanics.", "By applying tools from calculus and algebra, ornithologists transform raw tracking data into actionable insights—proving that behind every bird’s flight is a story written in equations."]