Why does my MacBook Pro randomly shut down?

I bought this computer in November 2009. About 2 months ago I upgraded the RAM and replaced the battery. Recently it has been shutting down at random times with no warning.

I've been using a program to monitor the CPU, battery, and other components' temperatures. The CPU has never gone above 70 C and usually remains at 45 C. The battery stays around 25 C.

I have removed and reinserted the battery and RAM. I blew out all the dust with a can of compressed air although there wasn't much since I had cleaned it out a few months ago.

After the computer shuts down, it usually is non-responsive to pressing the power button. Sometimes it will respond immediately, in a few minutes, or 15-20 minutes. When it is off, the power indicator button works fine and the correct amount of lights, light up.

Does anyone have any insight on what could be wrong and / or how to fix it?

UPDATE:

So after taking out the battery and re-inserting it again, I've noticed that it has stopped shutting off entirely, but instead acts as though the battery has died. My best guess is one of the wires one the battery is faulty.