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Home » Did You Recognize This Perfume? Authorities Claim It Was the Key to Catching the Deadliest Mass Murderer! - AMAZONAWS

Did You Recognize This Perfume? Authorities Claim It Was the Key to Catching the Deadliest Mass Murderer! - AMAZONAWS

Did You Recognize This Perfume? Authorities Claim It Was Key to Catching the Deadliest Mass Murderer

📅 March 6, 2026 👤 scraface
Mar 06, 2026
Did You Recognize This Perfume? Authorities Claim It Was the Key to Catching the Deadliest Mass Murderer!

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📌 Question: A science educator designs a workshop where each teacher contributes $ a $ lesson plans and $ b $ activities. If $ a + b = 12 $ and $ a^2 + b^2 = 90 $, find $ a^3 + b^3 $.
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📌 Using the dilution formula \( C_1V_1 = C_2V_2 \), where \( C_1 = 0.5 \) M, \( V_1 = 300 \) mL, \( V_2 = 500 \) mL, solve for \( C_2 \).
📌 Solution: Use partial fractions: $ \frac{1}{k(k+2)} = \frac{1}{2}\left( \frac{1}{k} - \frac{1}{k+2} \right) $. The sum becomes $ \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} - \sum_{k=1}^{50} \frac{1}{k+2} \right) = \frac{1}{2} \left( \sum_{k=1}^{50} \frac{1}{k} - \sum_{k=3}^{52} \frac{1}{k} \right) $. Telescoping gives $ \frac{1}{2} \left( \frac{1}{1} + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) = \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \cdot \frac{3975}{2652} = \frac{1325}{1768} $. \boxed{\dfrac{1325}{1768}}
📌 \text{Ratio} = \frac{\pi \cdot 6 \cdot 2\sqrt{3}}{864} = \frac{12\pi \sqrt{3}}{864} = \frac{\pi \sqrt{3}}{72}
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