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Home » Stop Mining Turbulence—Here’s the Quickest Way to Collect Obsidian! - go-checkin.com

Stop Mining Turbulence—Here’s the Quickest Way to Collect Obsidian! - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Final answer: $ oxed{rac{\pi c}{z + c}} $

📌 Question: An underwater archaeologist discovers a circular artifact partially submerged in water, forming a spherical cap with height $ h $ and base radius $ r $. If the full sphere from which the cap was formed has radius $ R $, what is the volume of the spherical cap in terms of $ h $, $ r $, and $ R $?

📌 Solution: The volume $ V $ of a spherical cap of height $ h $ cut from a sphere of radius $ R $ is given by the formula:

📌 V = rac{\pi h^2}{3}(3R - h)

📌 Alternatively, using the base radius $ r $, we know from geometry that $ r^2 = h(2R - h) $. However, the volume can be expressed directly in terms of $ h $ and $ R $. Substituting into the standard formula, we get:

📌 This expression gives the volume of the cap in terms of the height and sphere radius. Noting that $ r $ is not needed unless to relate to $ h $ and $ R $, but the volume in simplest form is:

📌 oxed{rac{\pi h^2}{3}(3R - h)}

📌 Question: A pharmacologist models the diffusion of a spherical drug particle in a gel medium. The particle has radius $ x $, and it interacts with a hemispherical receptor of radius $ 3x $. What is the ratio of the volume of the particle to the volume of the receptor?

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