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Home » where \( r \) is the radius of the base and \( h \) is the height. - Silent Sales Machine

where \( r \) is the radius of the base and \( h \) is the height. - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: Find the point on the line $ y = 2x + 1 $ closest to $ (3, 4) $.

📌 Solution: The closest point minimizes distance. Parametrize the line as $ (t, 2t + 1) $. The squared distance to $ (3, 4) $ is $ (t - 3)^2 + (2t + 1 - 4)^2 = (t - 3)^2 + (2t - 3)^2 $. Differentiate: $ 2(t - 3) + 8t - 12 = 0 \Rightarrow 10t - 18 = 0 \Rightarrow t = 1.8 $. Substituting, the point is $ (1.8, 4.6) $. oxed{(1.8,\ 4.6)}

📌 Question: Find the maximum value of $ (\sin x + 2\cos x)^2 + \sin^2 x $.

📌 Solution: Expand $ (\sin x + 2\cos x)^2 = \sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Add $ \sin^2 x $: total $ 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Simplify using identities: $ 2(1 - \cos^2 x) + 2\sin 2x + 4\cos^2 x = 2 + 2\cos^2 x + 2\sin 2x $. Let $ u = \cos^2 x $, $ \sin 2x = 2\sin x \cos x $. Alternatively, rewrite original expression as $ \sin^2 x + 4\sin x \cos x + 4\cos^2 x + \sin^2 x = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Let $ f(x) = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Use $ \sin^2 x = rac{1 - \cos 2x}{2} $, $ \cos^2 x = rac{1 + \cos 2x}{2} $, $ \sin x \cos x = rac{\sin 2x}{2} $:

📌 $ f(x) = 2\left(rac{1 - \cos 2x}{2}

📌 ight) + 4\left(rac{\sin 2x}{2}

📌 ight) + 4\left(rac{1 + \cos 2x}{2}

📌 ight) = (1 - \cos 2x) + 2\sin 2x + (2 + 2\cos 2x) = 3 + \cos 2x + 2\sin 2x $.

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