PATS Key/Transceiver

I broke my last set of keys for my car (2005 Ford Focus) I have the metal part of the key but the black part with the chip is gone. Had my car towed to a dealership cause they said they had to match the computer with the new key, this was monday evening. They told me my car would be ready by noon today (Tuesday). I call them at noon im told they dont have the key I need they are still trying to find one. I get called back at 3pm they say they couldnt find the key I need so they ordered it, they should have it by Wednesday and I can pick my car up Thursday after 12pm. Then I get a call at 5:30pm and they tell me my transceiver is not woking its going to cost $180 to replace and the key that they spent all day trying to find just poped up out of thin air. How likely is this and if the transceiver is the problem is $180 a fair price?