A cylinder is inscribed in a right circular cone of height 8 and radius (at the base) equal to 8. What are the dimensions of such a cylinder which has maximum volume?

Answer 1

Let r be the radius of the inscribed cylinder. The top edge of the cylinder (where the height is h) must touch the cone, where h and r are related by the straight-line equation
h = 8 - r.
The volume of the cone is
V (r) = pi*r^2 h = pi*r^2*(8-r)
= pi*[8 r^2 - r^3]

When the cylinder's volume is a maximum, dV/dr = 0, so
16r = 3r^2
r = 16/3, which is 2/3 the height of the come.