Question

a cone with height h is inscribed a larger cone with height 12 so that its vertex is at the center of the base of the larger cone. find height of inner cone that maximizes its volume

Answer 1

Use similar triangles to find r in terms R

Answer 2

OCD and OAB are similar triangles.
OC / OA = r / R
(12-h) / 12 = r / R
r = (12-h)/12 * R

Volume of inner cone V = 1/3 * pi * r^2 * h
r = (12-h)/12 * R Put this in V above and maximize V wrt h. (Find dV/dh and equate it to zero and solve for h).