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Home » Flowery Branches Unveiled – Transform Your Home with This Emotionally Charged Beauty! - Moon Smoking

Flowery Branches Unveiled – Transform Your Home with This Emotionally Charged Beauty! - Moon Smoking

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 \]Question: A cartographer measures the lower edge of a mountain ridge as a chord of a circular lake, finding it to be 10 km long, and the perpendicular distance from the lakeâs center to this chord as 3 km. What is the radius of the lake in kilometers?

📌 Solution: Let the radius of the circular lake be $ r $ km. The chord length is 10 km, so half the chord length is $ 5 $ km. The perpendicular distance from the center to the chord is 3 km. By the Pythagorean Theorem applied to the right triangle formed by the radius, half the chord, and the perpendicular from the center to the chord, we have:

📌 r^2 = 3^2 + 5^2 = 9 + 25 = 34

📌 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}

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