oxed{rac{\pi h^2}{3}(3R - h)}

February 22, 2026

$oxed{ rac{\pi h^2}{3}(3R - h)}$

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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:
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

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