Show that the volume of the largest right circular cylinder that can be inscribed in a given right circular cone is 4/9 the volume of the cone

Answer 1

draw a side view
If the cone has radius R and height H, and the cylinder has radius r and height h, then using similar triangles,
R/H = (R-r)/h
The volume of the cylinder is
v = πr^2 h = πr^2 * (R-r) * H/R = kr^2(R-r) for k=πH/R
dv/dr = 2Rr - 3r^2 = r(2R-3r)
dv/dr = 0 when r = 2/3 R
when r = 2/3 R, h = (R-r) * H/R = R/3 * H/R = H/3
So πr^2 h = π(2/3 R)^2*(H/3) = 4/9 * 1/3 πR^2 H