I have a poloroid 15" tla-01511c the picture comes on at first then goes black with just sound in about 30 seconds

Most likely the TV has a power supply related problem. The most common thing to go out is the circuit board that supply power to the screen. Commonly called a Backlight Inverter Board.
But, I can't say it's 100% the problem because I can't actually look at the tv over the internet.
If by chance the TV is still under warranty, go that route.
If not under warranty and If you have some electronic experience, if you can find and purchase a Service Manual over the internet; It will help you diagnose the source of the problem and fix it by soldering in the new part. The Service Manual will give you circuit diagrams, test points, and part# lists. Usually, cost under $20. And most cases the new part cost less than $15.

I hope I shed some light on the subject for you.