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Home » + 26 = 93 \text million miles - AMAZONAWS

+ 26 = 93 \text million miles - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Combine like terms:

📌 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:

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\boxed{13\sqrt{3} \text{ square centimeters}}
A right triangle has a hypotenuse of length \( z = 10 \) cm and an ins...
A_{\text{circle}} = \pi c^2 = \pi (2)^2 = 4\pi
For a right triangle, the radius \( c \) of the inscribed circle is gi...
where \( a \) and \( b \) are the legs. But we also know:
A_{\text{triangle}} = \frac{1}{2}ab = r \cdot s
where \( r = c = 2 \) and \( s = \frac{a + b + z}{2} \) is the semi-pe...
Using the identity for right triangles:
c = \frac{a + b - z}{2} \Rightarrow a + b = 2c + z = 4 + 10 = 14
Now, the semi-perimeter:

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