Question:** A zoologist observes that the population \( P(t) \) of a certain species in the Amazon rainforest follows the quadratic model \( P(t) = kt^2 + mt + n \). Given \( P(1) = 120 \), \( P(2) = 150 \), and \( P(3) = 210 \), find the coefficients \( k \), \( m \), and \( n \).

["Understanding the Population Growth of a Species Using Quadratic Modeling: A Zoologist’s Approach", "In ecological studies, modeling population dynamics is essential for conservation and understanding ecosystem health. In the Amazon rainforest, zoologists often use quadratic functions to describe population changes over time due to their ability to capture increasing or decelerating growth patterns. This article explores how a zoologist deduces the coefficients ( k ), ( m ), and ( n ) in a quadratic population model ( P(t) = kt^2 + mt + n ), using real field data from three observation periods.", "## The Study: Capturing Population Trends with a Quadratic Equation", "A recent study tracked a certain animal species in the Amazon with the following population data:\n- At time ( t = 1 ) month: ( P(1) = 120 )\n- At time ( t = 2 ) months: ( P(2) = 150 )\n- At time ( t = 3 ) months: ( P(3) = 210 )", "Assuming population growth follows the quadratic form ( P(t) = kt^2 + mt + n ), a zoologist seeks to determine the exact values of ( k ), ( m ), and ( n ) to improve predictive accuracy and inform conservation strategies.", "## Setting Up the System of Equations", "Using the given data points, substitute ( t ) and ( P(t) ) into the quadratic model:", "1. For ( t = 1 ):\n[\nk(1)^2 + m(1) + n = 120 \quad \Rightarrow \quad k + m + n = 120 \quad \ ext{(Equation 1)}\n]", "2. For ( t = 2 ):\n[\nk(2)^2 + m(2) + n = 150 \quad \Rightarrow \quad 4k + 2m + n = 150 \quad \ ext{(Equation 2)}\n]", "3. For ( t = 3 ):\n[\nk(3)^2 + m(3) + n = 210 \quad \Rightarrow \quad 9k + 3m + n = 210 \quad \ ext{(Equation 3)}\n]", "## Solving the System Step-by-Step", "We now solve the three linear equations:", "Step 1: Subtract Equation 1 from Equation 2:\n[\n(4k + 2m + n) - (k + m + n) = 150 - 120\n]\n[\n3k + m = 30 \quad \ ext{(Equation 4)}\n]", "Step 2: Subtract Equation 2 from Equation 3:\n[\n(9k + 3m + n) - (4k + 2m + n) = 210 - 150\n]\n[\n5k + m = 60 \quad \ ext{(Equation 5)}\n]", "Step 3: Subtract Equation 4 from Equation 5:\n[\n(5k + m) - (3k + m) = 60 - 30\n]\n[\n2k = 30 \quad \Rightarrow \quad k = 15\n]", "Step 4: Substitute ( k = 15 ) into Equation 4:\n[\n3(15) + m = 30 \quad \Rightarrow \quad 45 + m = 30 \quad \Rightarrow \quad m = -15\n]", "Step 5: Substitute ( k = 15 ) and ( m = -15 ) into Equation 1:\n[\n15 - 15 + n = 120 \quad \Rightarrow \quad n = 120\n]", "## The Final Model and Interpretation", "The quadratic model describing the population is:\n[\n\boxed{P(t) = 15t^2 - 15t + 120}\n]", "This model reveals:\n- The population grows rapidly over time (( k > 0 )), indicating accelerating growth in favorable conditions.\n- The linear coefficient ( m = -15 ) suggests a temporary decline in early months, potentially due to predation or resource scarcity.\n- At ( t = 0 ), the model predicts ( P(0) = 120 ), implying an initial population baseline.", "## Conclusion", "By applying algebraic techniques to field data, the zoologist successfully determined that the population follows ( P(t) = 15t^2 - 15t + 120 ). This precise model supports better ecological forecasting and targeted protection measures in the dynamic environment of the Amazon rainforest.", "Understanding population dynamics through structured mathematical modeling remains a cornerstone of modern zoology and conservation science."]