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Home » How Long to Drain 100 Liters at 15 L/h? - Dachbleche24

How Long to Drain 100 Liters at 15 L/h? - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solving equations 1 and 2: From 1, $ a = 5 - 3b $. Substituting into 2: $ 2(5 - 3b) + 4b = 6 \Rightarrow 10 - 6b + 4b = 6 \Rightarrow -2b = -4 \Rightarrow b = 2 $. Then $ a = 5 - 6 = -1 $.

📌 Solving equations 3 and 4: From 3, $ c = 7 - 3d $. Substituting into 4: $ 2(7 - 3d) + 4d = 8 \Rightarrow 14 - 6d + 4d = 8 \Rightarrow -2d = -6 \Rightarrow d = 3 $. Then $ c = 7 - 9 = -2 $.

📌 Thus, $ \mathbf{M} = \begin{pmatrix} -1 & 2 \\ -2 & 3 \end{pmatrix} $. Final answer: $ \boxed{\begin{pmatrix} -1 & 2 \\ -2 & 3 \end{pmatrix}} $.

📌 Question: Find $ y $ such that the vectors $ \begin{pmatrix} 2 \\ y \\ -1 \end{pmatrix} $ and $ \begin{pmatrix} 3 \\ -2 \\ y \end{pmatrix} $ are orthogonal.

📌 Solution: Two vectors are orthogonal if their dot product is zero. Compute:

📌 2(3) + y(-2) + (-1)(y) = 6 - 2y - y = 6 - 3y = 0.

📌 Solving $ 6 - 3y = 0 $ gives $ y = 2 $. Final answer: $ \boxed{2} $.

📌 Question: If $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vectors with $ \mathbf{a} \cdot \mathbf{b} = \frac{1}{2} $, $ \mathbf{b} \cdot \mathbf{c} = \frac{\sqrt{3}}{2} $, find the maximum value of $ \mathbf{a} \cdot \mathbf{c} $.

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