A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 7. What are the dimensions of such a cylinder which has maximum volume?

Answer 1

Center a cylinder of radius r on the axis of the cone of height H and base radius R. If the cylinder has height h, using similar triangles,

H/R = (H-h)/r
h = H - Hr/R

volume of cylinder is

v = πr²h = πr²(H - Hr/R)
= πHr² - πH/R r³

dv/dr = 2πHr - 3πH/R r²
= πHr(2 - 3r/R)

max at r = 2R/3, h = H/3

volume = πr²h = π(2R/3)²(H/3) = 4πR²H/27