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Home » \( x \approx 39 \) - High Altitude Science

\( x \approx 39 \) - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Solution: We use the well-known algebraic identity:

📌 a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).

📌 Given that $ a + b + c = 0 $, substitute into the identity:

📌 a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - bc - ca) = 0.

📌 a^3 + b^3 + c^3 = 3abc.

📌 This completes the proof.

📌 oxed{a^3 + b^3 + c^3 = 3abc}

📌 Question: Compute $ \sum_{k=1}^{50} rac{1}{k(k+2)} $.

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A rectangular garden measuring 12 meters by 8 meters is surrounded by...
Area of the garden = 12 meters × 8 meters = 96 square meters
Area of the garden and path = 200 square meters
Let the width of the path be x meters.
Total dimensions including path = (12 + 2x) meters × (8 + 2x) meters
Equation: (12 + 2x)(8 + 2x) = 200
Expanding: 96 + 24x + 16x + 4x^2 = 200
Simplifying: 4x^2 + 40x + 96 = 200
4x^2 + 40x - 104 = 0
Dividing by 4: x^2 + 10x - 26 = 0

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