Gwen just bought a bag of gumdrops! However, she does not like carrying gumdrops in plastic bags; instead, she wants to pack her gumdrops in a cylindrical tube of diameter d. Given that each of her gumdrops are perfect spheres of radii r1, r2, . . . , rn, find the shortest length tube Gwen can use to store her gumdrops.
Figure 4: Gumdrops packed into a cylindrical tube.
You should assume that the gumdrop radii are sufficiently large that no three gumdrops can be simultaneously in contact with each other while fitting in the tube. Given this restriction, it may be helpful to realize that the gumdrops will always be packed in such a way that their centers lie on a single two-dimensional plane containing the axis of rotation of the tube.