Originally posted by uzless
You have to show your work grasshopper....now, PAINT THE FENCE!
Allow me to wax-on, poetic...
Consider any two adjacent spheres in the cone. The small sphere has a radius "r" and the larger sphere has a radius "R". The ratio between the two radii is then simply R/r. Each pair of adjacent spheres exhibits the same ratio between radii since each pair must meet the same stipulations given in the question (touch each other, touch the sides of the cone). Therefore, to find the radius of the next sphere in line, we multiply the radius of the preceeding sphere by R/r. There are 5 spheres, so we have to scale up by this ratio 4 times to get the radius of the largest sphere, giving us the equation:
18 = 8 * (R/r)^4
(R/r)^4 = 18/8
(R/r) = (18/8)^(1/4) ~= 1.224744871
The middle sphere is scaled up twice from the original sphere, so its radius is 8 * (R/r)^2 = 12.
Alternatively, one could argue that since the radii form an ascending geometric sequence, the middle sphere must have a radius equal to the geometric mean of the two outside spheres:
R(mid) = SQRT(8*18) = SQRT(144) = 12.