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where to stay in boston university - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 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

📌 Now, the elevation at the centroid is a weighted average of the heights at $ A, B, C $, with weights equal to the areas of sub-triangles from centroid to each edge. Since centroid divides the triangle into three equal-area sub-triangles, the elevation at the centroid is the average of the $ z $-coordinates:

📌 \frac{a + b + c}{3} = 12

📌 Thus, the $ z $-coordinate of the centroid is consistent with the average elevation:

📌 \boxed{\frac{11}{3}}

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