Question

volume of largest right cylinder inscribed in a sphere of radius root3

Answer 1

Answer:

Volume of cylinder = pi * r^2 * h  where r = radius of cylinder and h the height

r = R cos theta where theta = angle between r and R at point of contact

V = pi * R^2 * cos^2 theta * 2 R sin theta   where h/2 = R sin theta

V = 2 * pi * R^3 cos^2 theta sin theta

dV / d theta = 2 * pi * R^3  * (-2 cos theta sin^2 theta + cos^3 theta) = 0

2 cos theta sin^2 theta = cos^3 theta

2 sin^2 theta = cos^2 theta

tan^2 theta= 1/2     theta = 35.3 deg     cos theta = .816  sin theta = .577

V = 2 * pi * 3 * 3^1/2 * .816^2 * .577 = 12.5