It sounds as though your computer is overheating and then shutting itself off to prevent damage. Many motherboards have that feature built in where it will monitor the temperature and then turn itself off when it gets to a certain level. Since your computer is going up to 93 degrees Fahrenheit it sounds like this is what is happening. You can check the BIOS to see what the shut off temperature is set at to verify that this is why it is shutting down.
The reason for it getting too hot is most likely a bad fan somewhere. If the processor fan is not working or not spinning fast enough then that will cause the processor to overheat really quickly. Some motherboards also have a setting to tell you the speed/RPM of the processor fan so you will know how fast it is spinning. I would open the case and then turn on the computer and check the processor and case fans to see if any are out. Also if you have a buildup of dust inside then that can cause things to get warm as well.