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Home » 25\pi x^2 + 65\pi x^2 = 90\pi x^2. - Abu Waleed Tea

25\pi x^2 + 65\pi x^2 = 90\pi x^2. - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 5Question: Find the point on the line $ y = -\frac{1}{2}x + 5 $ that is closest to the point $ (4, 3) $, modeling the optimal meeting spot between two social cliques.

📌 Solution: The closest point on a line to a given point is the perpendicular projection. The slope of the line is $ m = -\frac{1}{2} $, so the perpendicular slope is $ 2 $. The equation of the perpendicular line through $ (4, 3) $ is $ y - 3 = 2(x - 4) $. Solving the system:

📌 y = -\frac{1}{2}x + 5 \\

📌 y = 2x - 5

📌 Set equal: $ 2x - 5 = -\frac{1}{2}x + 5 $ → $ \frac{5}{2}x = 10 $ → $ x = 4 $. Substitute $ x = 4 $ into $ y = 2x - 5 $: $ y = 3 $. The closest point is $ (4, 3) $, which coincides with the given point (implying the point lies on the line). However, verifying: $ 3 = -\frac{1}{2}(4) + 5 = -2 + 5 = 3 $. Thus, the closest point is $ \boxed{(4, 3)} $.

📌 Question: A data scientist models urban traffic flow with a transformation matrix $ \mathbf{M} $ that rotates traffic patterns by $ 45^\circ $ and scales them by $ \sqrt{2} $. Find $ \mathbf{M} $.

📌 Solution: A rotation matrix for $ \theta = 45^\circ $ is $ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $. Scaling by $ \sqrt{2} $ multiplies each entry:

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