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Home » Alternatively, perhaps \( S_n = 210 \), but let’s accept the exact solution. - Baxtercollege

Alternatively, perhaps \( S_n = 210 \), but let’s accept the exact solution. - Baxtercollege

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 x^4 + ( -2x^3 + 2x^3 ) + (3x^2 - 4x^2 + 3x^2) + (6x - 6x) + 9 = x^4 + 2x^2 + 9

📌 Final answer: $ \boxed{x^4 + 2x^2 + 9} $Question: A cartographer is analyzing elevation data from a triangular region defined by points $ A = (1, 2), B = (4, 6), C = (7, 3) $. If the elevation at each vertex is proportional to its respective $ z $-coordinate, and the average elevation over the triangle is $ 12 $, what is the $ z $-coordinate of the centroid of triangle $ ABC $?

📌 Solution: The centroid $ G $ of triangle $ ABC $ is the average of its vertices' coordinates:

📌 G = \left( \frac{1 + 4 + 7}{3}, \frac{2 + 6 + 3}{3} \right) = \left( 4, \frac{11}{3} \right)

📌 The $ z $-coordinate of the centroid is $ \frac{11}{3} $. Since the average elevation is given as 12, and elevation is proportional to the $ z $-coordinate, we assume linear interpolation over the triangle. The centroid gives the average value of linear interpolation, so the elevation at the centroid equals the average elevation:

📌 \frac{z_A + z_B + z_C}{3} = 12

📌 Let $ z_A = a, z_B = b, z_C = c $. Then:

📌 \frac{a + b + c}{3} = 12 \Rightarrow a + b + c = 36

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Substitute \( a = 1 \) into Equation 4:
Substitute \( a = 1 \) and \( b = 2 \) into the first equation:
Thus, the values are \( a = 1 \), \( b = 2 \), \( c = 3 \).
\boxed{a = 1, \, b = 2, \, c = 3}
Question:** A zoologist observes that the population \( P(t) \) of a c...
\( P(1) = k(1)^2 + m(1) + n = 120 \implies k + m + n = 120 \)
\( P(2) = k(2)^2 + m(2) + n = 150 \implies 4k + 2m + n = 150 \)
\( P(3) = k(3)^2 + m(3) + n = 210 \implies 9k + 3m + n = 210 \)
Set up the system:
k + m + n &= 120 \quad \text{(1)}\\

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