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📰 Solution: The closest point is the projection of $(4, 3)$ onto the line. The formula for the projection of a point $(x_0, y_0)$ onto $ax + by + c = 0$ is used. Rewriting the line as $\frac{1}{2}x + y - 5 = 0$, we compute the projection. Alternatively, parametrize the line and minimize distance. Let $x = t$, then $y = -\frac{1}{2}t + 5$. The squared distance to $(4, 3)$ is $(t - 4)^2 + \left(-\frac{1}{2}t + 5 - 3\right)^2 = (t - 4)^2 + \left(-\frac{1}{2}t + 2\right)^2$. Expanding: $t^2 - 8t + 16 + \frac{1}{4}t^2 - 2t + 4 = \frac{5}{4}t^2 - 10t + 20$. Taking derivative and setting to zero: $\frac{5}{2}t - 10 = 0 \Rightarrow t = 4$. Substituting back, $y = -\frac{1}{2}(4) + 5 = 3$. Thus, the closest point is $(4, 3)$, which lies on the line. $\boxed{(4, 3)}$ 📰 Question: A hydrologist models groundwater flow with vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find the angle between these flow directions. 📰 Solution: The angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $2(1) + (-3)(4) = 2 - 12 = -10$. Compute magnitudes: $\|\mathbf{a}\| = \sqrt{2^2 + (-3)^2} = \sqrt{13}$, $\|\mathbf{b}\| = \sqrt{1^2 + 4^2} = \sqrt{17}$. Thus, $\cos\theta = \frac{-10}{\sqrt{13}\sqrt{17}}$. Rationalizing, $\theta = \arccos\left(-\frac{10}{\sqrt{221}}\right)$. $\boxed{\arccos\left(-\dfrac{10}{\sqrt{221}}\right)}$ 📰 Photobooth App 4208839 📰 The Conspiracy Against The Human Race A Contrivance Of Horror 9295558 📰 Get The Windows 10 Iso Download Nowfree File Included For Instant Setup 8022574 📰 Run 2 Temple Run 2 The Ultimate Feature Packed Adventure You Cant Miss 8076314 📰 Solution We Are To Find The Smallest Batch Size B Such That 6335172 📰 Get This Viral Guy Locked Animal In Your Brain Mind Blowing Meme Alert 5481263 📰 Adam Lamberts Bombshell Wife Revealedno One Saw This Coming 595631 📰 Indiana Connections Academy 6176024 📰 Watch Your Tax Bill Dropheres How To Navigate Long Term Capital Gains Brackets Like A Pro 2837485 📰 You Wont Guess The Hidden Superpowers Behind Everyones Favorite Robot Characters 5173842 📰 Shocked To See Carrier Stock Jumping Heres Whats Driving It 3638884 📰 This Secret Hex Editor Hack For Mac Will Transform Your File Editing Game 2113749 📰 Nahuatl Translator 2790355 📰 Dont Miss This Fidelity Roth Roth Solo 401K Strategy Thats Outperforming Everyone Else 7843751 📰 Homosexuality In The Bible 403184