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Home » \(a_2 = 3(2) + 1 = 7\). - Portal da AcÃºstica

\(a_2 = 3(2) + 1 = 7\). - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: We model the triangular grid using a coordinate system adapted to equilateral triangles, but the problem simplifies: we interpret lattice points inside a large equilateral triangle of side length 6 in a triangular lattice where vertices are at lattice points and side length corresponds to 6 units. However, for computational geometry in such grids, the classic approach uses area and Pick’s Theorem.

📌 Pick’s Theorem states:

📌 A = I + \frac{B}{2} - 1

📌 where $A$ is area, $I$ is interior lattice points, and $B$ is boundary lattice points. We seek $I$, but the problem wants interior points only. First, compute the area of the large equilateral triangle with side length $6$:

📌 A = \frac{\sqrt{3}}{4} \cdot 6^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}

📌 Now, determine $B$: the number of lattice points on the boundary. Each side has $6 + 1 = 7$ points, but corners are shared. Since it’s a triangle,

📌 B = 3 \cdot 7 - 3 = 21 - 3 = 18

📌 Now apply Pick’s Theorem:

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Calculate stepwise:
Year 1: 12000 × 0.88 = <<12000*0.88=10560>>10560
Year 2: 10560 × 0.88 = <<10560*0.88=9292.8>>9292.8
Year 3: 9292.8 × 0.88 = <<9292.8*0.88=8177.664>>8177.664
Year 4: 8177.664 × 0.88 = <<8177.664*0.88=7196.34432>>7196.34432
Year 5: 7196.34432 × 0.88 ≈ <<7196.34432*0.88=6332.836>>6332.836 → rou...
#### 6333
A linguist records that in a border town, 78% of people speak English,...
English speakers: 0.78 × 1500 = <<0.78*1500=1170>>1170.
French speakers: 0.52 × 1500 = <<0.52*1500=780>>780.

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