A GPS-tagged falcon’s altitude during dive follows the function h(t) = -5t² + 100t + 1500, where t is seconds. What is the maximum altitude, in meters, reached during the dive?
["Title: Understanding Falcon Dive Altitude: A Mathematical Model Explained", "Meta Description: Explore how GPS data reveals a falcon’s flight trajectory—specifically, how the altitude h(t) = -5t² + 100t + 1500 models its dive. Discover the maximum altitude reached and the science behind this graphical formulation.", "---", "When a GPS-tagged falcon dives, its altitude over time follows a precise mathematical pattern. Thanks to modern tracking technology, scientists and nature enthusiasts can now observe these astonishing maneuvers in real time—translating raw data into meaningful insights. One common model used to describe such dives is the quadratic function:", "[ h(t) = -5t^2 + 100t + 1500 ]", "where ( t ) represents time in seconds, and ( h(t) ) gives the falcon’s altitude in meters.", "### What Does This Equation Represent?", "This quadratic function describes parabolic motion—typical in objects moving under gravity with no horizontal acceleration, such as a falcon diving through the air. The negative quadratic term (( -5t^2 )) accounts for gravitational pull, gradually reducing altitude, while the linear term (( 100t )) reflects the initial upward or downward velocity during the dive.", "### Finding the Maximum Altitude", "Since the coefficient of ( t^2 ) is negative, the parabola opens downward, meaning it peaks at its vertex—the moment the falcon reaches maximum altitude.", "For any quadratic equation in the form:", "[ h(t) = at^2 + bt + c ]", "the time ( t ) at the vertex (and thus maximum altitude) is given by:", "[ t_{\ ext{max}} = -\frac{b}{2a} ]", "In our function:\n( a = -5 ), ( b = 100 )", "So,", "[\nt_{\ ext{max}} = -\frac{100}{2 \ imes (-5)} = \frac{100}{10} = 10 \ ext{ seconds}\n]", "Now substitute ( t = 10 ) into ( h(t) ) to find the maximum altitude:", "[\nh(10) = -5(10)^2 + 100(10) + 1500\n]\n[\nh(10) = -5(100) + 1000 + 1500\n]\n[\nh(10) = -500 + 1000 + 1500 = 2000 \ ext{ meters}\n]", "### Conclusion: Peak Performance at 2000 Meters", "The falcon reaches its highest point at 10 seconds into the dive, soaring to 2000 meters above ground level. This peak illustrates the dramatic speed and aerodynamic precision of applying gravity during a controlled dive—validated and quantified through GPS tracking and mathematical modeling.", "Understanding such altitude dynamics helps biologists study falcon behavior, optimize tracking technology, and even inspire innovations in drone design and aircraft safety.", "---", "Keywords: falcon dive altitude, GPS-tagged falcon, quadratic function altitude model, maximum altitude falcon, h(t) = -5t² + 100t + 1500, high-altitude falconry, flight dynamics, parabolic motion in nature, falcon behavior analysis, altitude optimization in aviation", "---", "Final Answer:\nThe maximum altitude reached during the dive is 2000 meters."]
