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Home » This Ant with a Bag Changed Everything—Shocked Scientists Forever! - Leaselab

This Ant with a Bag Changed Everything—Shocked Scientists Forever! - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Solution: Square both sides: $ x^2 + 2 + \frac{1}{x^2} = 25 \Rightarrow x^2 + \frac{1}{x^2} = 23 $. Multiply by 2: $ 2x^2 + \frac{2}{x^2} = 46 $. Final answer: $ \boxed{46} $.Question: In a sustainable desert agriculture project, the combined water efficiency of two systems is modeled by $ a + b = 10 $ and $ a^2 + b^2 = 58 $. Determine the total water efficiency cubed, $ a^3 + b^3 $.

📌 Solution: We start with the given equations $ a + b = 10 $ and $ a^2 + b^2 = 58 $. First, we compute $ ab $ using the identity $ (a + b)^2 = a^2 + 2ab + b^2 $. Substituting, $ 100 = 58 + 2ab $, so $ ab = 21 $. Next, we use the identity for $ a^3 + b^3 $: $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $. Plugging in the values: $ a^3 + b^3 = 10^3 - 3 \times 21 \times 10 = 1000 - 630 = 370 $. Thus, the total water efficiency cubed is $ \boxed{370} $.

📌 Question: An organic chemist synthesizes a compound with molecular formula $ C_nH_{2n} $, where the ratio of carbon to hydrogen atoms satisfies $ \frac{a}{b} = 3 $ and their sum is $ a + b = 16 $. Find the value of $ a^2 - b^2 $.

📌 Solution: Given $ \frac{a}{b} = 3 $, let $ a = 3b $. Substituting into $ a + b = 16 $: $ 3b + b = 16 $, so $ b = 4 $ and $ a = 12 $. Now, $ a^2 - b^2 $ factors as $ (a - b)(a + b) = (12 - 4)(16) = 8 \times 16 = 128 $. Therefore, the value is $ \boxed{128} $.

📌 Question: A retired engineer designs a rectangular

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