What do you have to dilated...

What do you have to be dilated for the hospital to go ahead and induce you? I'm 39weeks dilated to a 3, 60% thinned and I've been having really bad back contractions today and pretty consistent mild period cramps in the lower part of my belly and sometimes above my belly button. I'm a FTM so I can't really tell if they are BH or real contractions but some of the back contractions can take my breath away