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Home » Area of one equilateral triangle: \(\frac\sqrt34 s^2 = \frac\sqrt34 \times 36 = 9\sqrt3\). - AMAZONAWS

Area of one equilateral triangle: \(\frac\sqrt34 s^2 = \frac\sqrt34 \times 36 = 9\sqrt3\). - AMAZONAWS

📅 March 6, 2026 👤 scraface
Mar 06, 2026
Area of one equilateral triangle: \(\frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}\).

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📌 Solution: Complete the square for $x$ and $y$. For $x$: $9(x^2 - 2x) = 9[(x - 1)^2 - 1] = 9(x - 1)^2 - 9$. For $y$: $-16(y^2 - 4y) = -16[(y - 2)^2 - 4] = -16(y - 2)^2 + 64$. Substitute back: $9(x - 1)^2 - 9 - 16(y - 2)^2 + 64 = 144$. Simplify: $9(x - 1)^2 - 16(y - 2)^2 = 89$. The center is at $(1, 2)$. Thus, the center is $oxed{(1, 2)}$.
📌 Thus, the smallest such integer is $\boxed{5}$.Question: An entomologist studies 5 species of bees, 3 species of wasps, and 2 species of beetles. If she observes one species per day for 10 days, how many distinct observation sequences are possible if species within each category are indistinguishable?

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