A meteorologist models atmospheric pressure variation with altitude using the function \( P(h) = ah^2 + bh + c \). Given \( P(0) = 1013 \), \( P(1) = 1000 \), and \( P(2) = 960 \), find the value of \( a + b + c \).

["Modeling Atmospheric Pressure with Altitude: A Mathematical Approach Using Quadratic Functions", "Understanding how atmospheric pressure changes with altitude is crucial in meteorology, aviation, and environmental science. Atmospheric pressure decreases as elevation increases—a predictable pattern often modeled using quadratic functions for simplicity and accuracy in moderate altitudes. In this article, we explore how a meteorologist can model pressure variation using the quadratic function ( P(h) = ah^2 + bh + c ), using real-world data at three key altitudes to determine the precise coefficients and compute ( a + b + c ), a key sum revealing baseline pressure parameters.", "---", "### The Pressure-Altitude Relationship", "The International Standard Atmosphere model shows that pressure decreases roughly exponentially, but for many practical short-range applications, a quadratic approximation offers both simplicity and insight. Here, we fit a quadratic function ( P(h) = ah^2 + bh + c ) to three elevation-pressure pairs:", "- At sea level (( h = 0 )): ( P(0) = 1013 ) hPa\n- At 1,000 meters (( h = 1 )): ( P(1) = 1000 ) hPa\n- At 2,000 meters (( h = 2 )): ( P(2) = 960 ) hPa", "These points form a discrete dataset enabling precise parameter estimation.", "---", "### Step 1: Use the given data to form a system of equations", "Given ( P(h) = ah^2 + bh + c ), substitute the known values:", "1. ( P(0) = 1013 \Rightarrow a(0)^2 + b(0) + c = 1013 \Rightarrow c = 1013 )\n2. ( P(1) = 1000 \Rightarrow a(1)^2 + b(1) + c = 1000 \Rightarrow a + b + c = 1000 )\n3. ( P(2) = 960 \Rightarrow a(2)^2 + b(2) + c = 960 \Rightarrow 4a + 2b + c = 960 )", "---", "### Step 2: Substitute ( c = 1013 ) into other equations", "From equation (2):\n( a + b + 1013 = 1000 \Rightarrow a + b = -13 )  (Equation A)", "From equation (3):\n( 4a + 2b + 1013 = 960 \Rightarrow 4a + 2b = -53 )  (Equation B)", "---", "### Step 3: Solve the system of two equations", "From Equation A: ( b = -13 - a )", "Substitute into Equation B:\n( 4a + 2(-13 - a) = -53 )\n( 4a - 26 - 2a = -53 )\n( 2a - 26 = -53 )\n( 2a = -27 \Rightarrow a = -13.5 )", "Now substitute back to find ( b ):\n( b = -13 - (-13.5) = 0.5 )", "We already know ( c = 1013 )", "---", "### Step 4: Compute ( a + b + c )", "[\na + b + c = -13.5 + 0.5 + 1013 = 1000\n]", "---", "### Why This Matters: The Sum ( a + b + c )", "While ( a + b + c = P(0) ) when ( h = 0 ), this sum also reflects the initial atmospheric baseline adjusted by the quadratic model parameters. In meteorological practice, calculating such combinations helps verify consistency, calibrate sensors, and interpret how deviations from ideal pressure with height manifest in real-world data.", "Moreover, although ( a + b + c = 1000 ), the function accurately reflects sub-linear pressure decay—highlighting how quadratic models balance simplicity and precision for near-ground atmospheric studies.", "---", "### Conclusion", "By fitting a quadratic model ( P(h) = ah^2 + bh + c ) to measured atmospheric pressure at three altitudes, we determined the coefficients with high accuracy. The computation reveals that ( a + b + c = 1000 ), matching the sea-level pressure and illustrating how mathematical modeling supports precise environmental predictions.", "For meteorologists, engineers, and scientists, such models are foundational tools—turning numbers into meaningful insight about Earth’s dynamic atmosphere.", "---", "Keywords: atmospheric pressure model, altitude and pressure, quadratic function P(h), meteorology, quadratic fitting, sea level pressure, aerodynamics, pressure variation, mathematical modeling in meteorology, altitude equation, ( P(h) = ah^2 + bh + c ), ( a + b + c )", "Meta Description: A practical guide to modeling atmospheric pressure variation with altitude using a quadratic function. Learn how a meteorologist uses real data (P(0)=1013, P(1)=1000, P(2)=960) to solve for coefficients and compute ( a + b + c = 1000 )."]