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hilton garden inn ontario rancho cucamonga - Anadea

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 $ (25, 0) $: $ x = \pm5 $, $ y = 0 $ → 2 solutions

📌 Total solutions: $ 2 + 4 + 4 + 2 = 12 $.

📌 \boxed{12}

📌 Question: An ornithologist tracks a bird’s flight path, which follows a parabolic trajectory modeled by $ y = -x^2 + 6x + 1 $. Determine the maximum height reached by the bird.

📌 Solution: The maximum height of a parabola $ y = ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $. Here, $ a = -1 $, $ b = 6 $, so $ x = -\frac{6}{2(-1)} = 3 $. Substituting $ x = 3 $ into the equation:

📌 $ y = -(3)^2 + 6(3) + 1 = -9 + 18 + 1 = 10 $.

📌 The maximum height is $ \boxed{10} $.

📌 Question: A museum curator catalogs a 3D artifact with three vertices of a regular octahedron at $ (1, 0, 0) $, $ (-1, 0, 0) $, and $ (0, 1, 0) $. Find the coordinates of the fourth vertex, given all coordinates are integers.

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