The neck part of the cut-out bottle is used for the nose of the rocket. 
 
Glueing technique for both the skirt and the nose cone: 
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Put some glue on the contact line. 
spread it with a popsicle stick.. or else.. 
then rotate the nose or skirt around a bit to further  spread the glue. 
Do not try to push it down too hard or it will tend to pop-out again ! 
you will notice how the fin/body ribs interlock ..Amazing..Ehh !
almost like if these bottles had been designed for water rocketry :-)
 
 if you want to fill the nose with water for added weight.(Increase the altitude)  
you will have to fill the gaps between the bottom ribs as well. 
with some more glue and a fine cartdrige nozzle. 
Note; 
(Use a glue that is not water based .Unlike some new "No More Nails"