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Home » The time for the first part is \( \frac15050 = 3 \) hours. - Groen Casting

The time for the first part is \( \frac15050 = 3 \) hours. - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 ight) = rac{18}{16} - rac{144}{256} = rac{9}{8} - rac{9}{16} = rac{18 - 9}{16} = rac{9}{16} = 0.5625

📌 So maximum of $ \sin 3x \sin x $ is $ rac{9}{16} $, thus minimum of $ f(x) = 1 - rac{9}{16} = rac{7}{16} $? But wait â this contradicts earlier thought.

📌 But where: $ f(x) = 1 - \sin 3x \sin x $, so minimum of $ f $ is when $ \sin 3x \sin x $ is **maximized**, which is $ rac{9}{16} $, so:

📌 f_{\min} = 1 - rac{9}{16} = rac{7}{16}

📌 But earlier we saw $ f(x) = 1 - rac{1}{2}(\cos 2x - \cos 4x) $. Letâs test $ x = rac{\pi}{2} $:

📌 $ \sin^2 rac{\pi}{2} = 1 $, $ \cos^2(2 \cdot rac{\pi}{2}) = \cos^2 \pi = (-1)^2 = 1 $, so $ f = 1 + 1 = 2 $? No â wait:

📌 Wait: $ \cos(2x) = \cos \pi = -1 $, $ \cos^2(2x) = 1 $, $ \sin^2 x = 1 $, so $ f(x) = 1 + 1 = 2 $? But that canât be â maximum of each is 1, but sum could be 2? But letâs compute $ f(x) = \sin^2 x + \cos^2(2x) \leq 1 + 1 = 2 $, but is $ f(x) = 2 $ possible? Only if $ \sin^2 x = 1 $ and $ \cos^2(2x) = 1 $.

📌 $ \sin^2 x = 1 \Rightarrow x = rac{\pi}{2} + k\pi $. At $ x = rac{\pi}{2} $, $ 2x = \pi $, $ \cos \pi = -1 $, $ \cos^2 \pi = 1 $. So yes: $ f\left(rac{\pi}{2}

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