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Home » Approximiere \( \sqrt37 \approx 6.08 \), also: - High Altitude Science

Approximiere \( \sqrt37 \approx 6.08 \), also: - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Solution: Apply De Moivre’s theorem: $(\cos 15^\circ + i \sin 15^\circ)^{12} = \cos(180^\circ) + i \sin(180^\circ) = -1 + 0i$.

📌 Question: If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors representing evolutionary traits, find the maximum value of $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ under positive scalar constraints.

📌 Solution: The scalar triple product equals the volume of the parallelepiped. Maximum occurs when vectors are orthogonal, giving $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = 1 \cdot 1 \cdot 1 \cdot \sin(90^\circ)\cos(0^\circ) = 1$.

📌 \boxed{1}Question: A research scientist is preparing 7 different chemical solutions. How many ways can they distribute these to 3 identical storage vials, ensuring each vial contains at least one solution?

📌 Solution: The problem requires partitioning 7 distinguishable solutions into 3 indistinguishable non-empty vials. This is equivalent to the Stirling numbers of the second kind, $ S(7, 3) $. The formula for Stirling numbers is:

📌 S(n, k) = \frac{1}{k!} \sum_{i=0}^k (-1)^{k-i} \binom{k}{i} i^n}

📌 Plugging in $ n = 7 $, $ k = 3 $:

📌 S(7, 3) = \frac{1}{6} \left( 3^7 - 3 \cdot 2^7 + 3 \cdot 1^7 \right) = \frac{1}{6} (2187 - 3 \cdot 128 + 3) = \frac{1}{6} (2187 - 384 + 3) = \frac{1806}{6} = 301

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