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ight)^2 - 12\left(rac32 - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: Let $ \vec{OA} = \mathbf{a} $, $ \vec{OB} = \mathbf{b} $. Then $ \overrightarrow{OC} = m\mathbf{a} + n\mathbf{b} $. For $ \overrightarrow{OC} \perp (\mathbf{a} - \mathbf{b}) $, their dot product is zero: $ (m\mathbf{a} + n\mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0 $. Expand: $ m\|\mathbf{a}\|^2 - m\mathbf{a} \cdot \mathbf{b} + n\mathbf{b} \cdot \mathbf{a} - n\|\mathbf{b}\|^2 = 0 $. Substitute $ \|\mathbf{a}\| = 2 $, $ \|\mathbf{b}\| = 3 $, $ \mathbf{a} \cdot \mathbf{b} = 2 \cdot 3 \cdot \cos 60^\circ = 3 $: $ 4m - 3m + 3n - 9n = 0 \Rightarrow m - 6n = 0 $. Choose $ n = 1 $, then $ m = 6 $. Normalize if needed, but the relation is $ m = 6n $. For simplicity, take $ n = 1 $, $ m = 6 $.

📌 \boxed{(6, 1)}Question: A hydrologist studying groundwater flow measures the volume of water (in liters) passing through a sensor at different times, recording values that are positive multiples of 5. If the cube of one such measurement is less than 1500, what is the greatest possible value of that measurement?

📌 Solution: Let the measurement be $ u $, a positive multiple of 5 such that $ u^3 < 1500 $.

📌 We test successive multiples of 5:

📌 $ u = 5 $: $ 5^3 = 125 $

📌 $ u = 10 $: $ 10^3 = 1000 $

📌 $ u = 15 $: $ 15^3 = 3375 $, which exceeds 1500.

📌 Thus, the greatest multiple of 5 satisfying $ u^3 < 1500 $ is $ u = 10 $.

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