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Home » J: Newtonâs generalized binomial theorem - Silent Sales Machine

J: Newtonâs generalized binomial theorem - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 r = \sqrt{34}

📌 The radius of the lake is $ oxed{\sqrt{34}} $ km.

📌 Question: An astronomer observes a protoplanetary disk rotating uniformly, where a planetesimal orbits along a circular path of radius $ R $. If the arc length between two opposite points on the orbit is $ 120^\circ $ of the circle, what is the ratio of the chord length (distance between these two points) to the radius $ R $?

📌 Solution: The central angle corresponding to the arc is $ 120^\circ $, or $ rac{2\pi}{3} $ radians. The chord length $ c $ subtended by a central angle $ heta $ in a circle of radius $ R $ is given by:

📌 c = 2R \sin\left(rac{ heta}{2}

📌 Substituting $ heta = rac{2\pi}{3} $:

📌 c = 2R \sin\left(rac{\pi}{3}

📌 ight) = 2R \cdot rac{\sqrt{3}}{2} = R\sqrt{3}

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