Thus, the perimeter is \(\boxed{30 + 10\pi \text{ meters}}\).Question: A palynologist observes that the number of pollen grains in a sample grows quadratically over time. If on day 2 there are 120 grains, on day 4 there are 360 grains, and on day 6 there are 720 grains, find the quadratic polynomial $ p(t) $ that models the number of pollen grains on day $ t $.

["Title: Quadratic Model for Pollen Grain Growth: Finding $ p(t) $ Given Daily Counts", "---", "Blog Post:", "When studying microscopic biological accumulations like pollen grains, palynologists often seek mathematical models to describe their growth patterns. Unlike linear or exponential models, quadratic models capture curvilinear growth — a common real-world phenomenon in biological systems.", "A recent observation reveals that the number of pollen grains in a sample increases quadratically over time:", "- On day ( t = 2 ), ( p(2) = 120 )\n- On day ( t = 4 ), ( p(4) = 360 )\n- On day ( t = 6 ), ( p(6) = 720 )", "Given these data points, we aim to find the quadratic polynomial ( p(t) = at^2 + bt + c ) that accurately models the pollen count as a function of time.", "---", "### Step 1: Set up equations using known values", "Substitute ( t = 2, 4, 6 ) into ( p(t) = at^2 + bt + c ):", "1. ( p(2) = 4a + 2b + c = 120 )\n2. ( p(4) = 16a + 4b + c = 360 )\n3. ( p(6) = 36a + 6b + c = 720 )", "---", "### Step 2: Solve the system of equations", "Subtract equation (1) from equation (2):", "[\n(16a + 4b + c) - (4a + 2b + c) = 360 - 120 \Rightarrow 12a + 2b = 240\n]\nSimplify:\n[\n6a + b = 120 \quad \ ext{(Equation A)}\n]", "Subtract equation (2) from equation (3):", "[\n(36a + 6b + c) - (16a + 4b + c) = 720 - 360 \Rightarrow 20a + 2b = 360\n]\nSimplify:\n[\n10a + b = 180 \quad \ ext{(Equation B)}\n]", "Now subtract Equation A from Equation B:", "[\n(10a + b) - (6a + b) = 180 - 120 \Rightarrow 4a = 60 \Rightarrow a = 15\n]", "Substitute ( a = 15 ) into Equation A:", "[\n6(15) + b = 120 \Rightarrow 90 + b = 120 \Rightarrow b = 30\n]", "Finally, substitute ( a = 15 ), ( b = 30 ) into equation (1):", "[\n4(15) + 2(30) + c = 120 \Rightarrow 60 + 60 + c = 120 \Rightarrow c = 0\n]", "---", "### Step 3: Write the final polynomial", "Thus, the quadratic polynomial modeling the number of pollen grains on day ( t ) is:", "[\np(t) = 15t^2 + 30t\n]", "This can also be factored as:", "[\np(t) = 15t(t + 2)\n]", "---", "### Step 4: Compute the perimeter-like growth expression — Interpreting the original statement", "The phrase "the perimeter is (\boxed{30 + 10\pi \ ext{ meters}})" seems metaphorical or misphrased in context — perimeter applies to two-dimensional shapes, whereas pollen grains grow in time and quantity, fitting a quadratic temporal model. However, it may symbolically emphasize the "boundary" or limit of growth under quadratic acceleration.", "But mathematically, we confirmed the growth model:", "[\n\boxed{p(t) = 15t^2 + 30t}\n]", "This model predicts increasing grain counts at an accelerating rate typical in biological aggregation processes.", "---", "Conclusion:\nBy fitting a quadratic polynomial to precise time-sampled data, palynologists gain a powerful tool for forecasting pollen accumulation — vital for paleoenvironmental reconstructions and allergy forecasting. The derived model ( p(t) = 15t^2 + 30t ) supports accurate, time-dependent predictions with a clear mathematical foundation.", "---", "Keywords: Palynology, pollen growth, quadratic model, quadratic polynomial, time-dependent growth, polynomial learning, palynologist, quadratic growth rate, pollen grain modeling\nMeta Description: A quadratic polynomial ( p(t) = 15t^2 + 30t ) models pollen grain count over days, derived from data at ( t = 2, 4, 6 ). Ideal for predicting biological accumulation trends.", "---", "References:\n- Time-series biological data modeling\n- Quadratic regression techniques in microparticle analysis\n- Applied algebra in palynological research"]