A young woman and her boyfriend plan to elope, but she must rescue him from his mother who has locked him in his room. The young woman has placed a $20$ foot long ladder against his house and is knocking on his window when his mother begins pulling the bottom of the ladder away from the house at a rate of $3$ feet per second. How fast is the top of the ladder $($and the young couple$)$ falling when the bottom of the ladder is $($a$)12$ feet from the bottom of the wall? $($b$)16$ feet from the bottom of the wall? $($c$)19$ feet from the bottom of the wall?