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Home » The number of such surjective functions is given by: - Veritas Home Health

The number of such surjective functions is given by: - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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?

📌 Solution: The volume of a sphere of radius $ x $ is $ V_{ ext{particle}} = rac{4}{3}\pi x^3 $. The volume of a hemisphere of radius $ 3x $ is half the volume of a full sphere:

📌 V_{ ext{receptor}} = rac{1}{2} \cdot rac{4}{3}\pi (3x)^3 = rac{2}{3}\pi \cdot 27x^3 = 18\pi x^3

📌 Now compute the ratio:

📌 rac{V_{ ext{particle}}}{V_{ ext{receptor}}} = rac{rac{4}{3}\pi x^3}{18\pi x^3} = rac{4}{3} \cdot rac{1}{18} = rac{4}{54} = rac{2}{27}

📌 Final answer: $ oxed{rac{2}{27}} $

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