A wildlife biologist models the migration distance (in kilometers) of a shorebird using the equation d = 50t + 12t², where t is time in days. What is the average speed, in km/day, between day 2 and day 5?
["Modeling Shorebird Migration: Understanding Daily Movement with a Biologist’s Equation", "Understanding how shorebirds migrate across vast distances is crucial for conservation and ecological research. A wildlife biologist recently developed a mathematical model to estimate the migration distance of a key shorebird species using the equation:", "d = 50t + 12t²,", "where d is the migration distance in kilometers and t is time in days. This model provides valuable insights into how fast these birds move over time, especially during critical phases of migration.", "### Calculating Migration Distance at Key Times", "To determine the average speed between day 2 and day 5, we first compute the migration distance at both time points:", "- At t = 2 days:\n d₂ = 50(2) + 12(2)² = 100 + 12×4 = 100 + 48 = 148 km", "- At t = 5 days:\n d₅ = 50(5) + 12(5)² = 250 + 12×25 = 250 + 300 = 550 km", "### Finding Average Speed Over the Interval", "Average speed is calculated as total distance traveled divided by total time:", "[\n\ ext{Average speed} = \frac{d_5 - d_2}{5 - 2} = \frac{550,\ ext{km} - 148,\ ext{km}}{3,\ ext{days}} = \frac{402,\ ext{km}}{3} = 134,\ ext{km/day}\n]", "### Biological Significance and Research Insight", "This average speed of 134 km/day reveals that the modeled shorebird covers approximately 130 kilometers per day during this phase of migration—remarkably consistent with observed patterns in similar species. Such modeling helps biologists track energy demands, timing of rest stops, and vulnerability to habitat loss. The quadratic term in the equation (12t²) reflects increasing daily distances likely due to reduced fat reserves, improved navigation, or favorable tailwinds.", "### Conclusion", "Using the wildlife biologist’s model d = 50t + 12t², the average speed of a shorebird migrating between day 2 and day 5 is 134 km/day. This precise mathematical approach supports deeper understanding of avian migration ecology and informs conservation strategies aimed at protecting critical stopover sites along these impressive journeys.", "---", "By combining field biology with predictive equations, researchers continue to uncover Nature’s patterns—one kilometer at a time."]
