I would suggest you see if the computer is dirty inside. This includes it's Power Supply.
The hardware components inside that desktop computer are cooled by air, are cooled by the Cooling Components.
Air is drawn into the computer case into the Top, or Side, (Depending on which way the computer is sitting), by a computer case fan, (Chassis Fan), and is exhausted out of the rear of the case by the Power Supply's fan..
As the drawn air makes it's way through the computer case, it is also drawn in by the Processor Fan.
The Processor Fan pushes air through a Heatsink, which sits on top of the Processor's case.
The Processor Fan, and Heatsink strive to keep the Processor within a certain operating temperature.
If the temperature is exceeded, the Processor turns off. (BIOS turns it off)
This is a Fail safe feature built-in to keep the Processor from burning up.
Typical construction of a Heatsink is plate of metal with tall, thin fins protruding from it.
The plate of metal absorbs heat from whatever object it is placed against, (In this case a Processor), and the tall, thin fins absorb heat from the plate of metal.
The tall, thin fins then radiate the heat away. The Processor Fan helps to carry heat away from the fins, by the produced air flow from the fan.
If the Chassis Fan, (Computer case fan), Processor Fan, and Heatsink are clogged with 'Gunk', the cooling capacity of these cooling components drops tremendously.
('Gunk' = Dirt, dust, hair, etc.)
I would start with cleaning out the inside of the computer, and the inside of the Power Supply.
Computer on a table, computer UNPLUGGED from power, computer case open.
Observe Anti-Static Precautions, and use a can of compressed air for computers to clean.
When you set the can of compressed air up, you will insert the plastic 'straw' into the nozzle, and break the lock tab used for shipping, off of the top of the can's nozzle.
Start at the inside top of the computer case, and start from one side. Left or Right is totally up to you.
Squeeze the trigger in all the way, go across the inside of the computer at the top.
As you arrive to the other side, let go of the trigger.
You are using the can of compressed air like a spray painter would.
Now drop down a space, and go across again. Spacing is up to you. You will see how much the air spray covers, and can judge accordingly.
Keep dropping down in rows until you arrive at the Processor Fan, and Heatsink.
The Processor Fan is only designed to spin so fast. Spinning it faster than it was designed for, can lead to premature failure of the fan's bearings.
For this reason i advise using Short Bursts.
Squeeze the trigger in all the way, but let go quickly.
(Remember what I stated about observing Anti-Static Precautions. If you are not aware of what Anti-Static Precautions are, please state in a Comment)
You may find that the buildup of 'Gunk' requires the use of Q-tips to help.
The Q-tips are just used to 'break the surface' of the Gunk. Not really used to remove it. Once the 'crust' of the gunk has been disturbed, the air can remove it more easily.
Use the straw to clean out in-between the fins of the Heatsink.
Air needs to flow freely in-between the fins.
Keep dropping down in rows now, until you reach the bottom of the computer case.
Hold the computer case with one hand, and the compressed air with the other hand.
Tilt the top of the case slightly towards you.
This helps to allow the air flow from the can of compressed air, to move the last of the Gunk out of the computer case.
Now go to the Power Supply's case from the inside of the computer.
Insert the tip of the straw part way into a ventilation hole, of the Power Supply's case.
HOLD on to the can of air with one hand, and the straw with the other hand.
The straw might shoot off of the nozzle otherwise, and it's NO fun trying to dig the straw out of the Power Supply's case.
Move from ventilation hole to ventilation hole, and spray air. You are just trying to clean the best you can. It won't be sparkling clean when you are done, but a LOT better than when you started.
Now go to the outside of the computer, and to the Power Supply fan.
Clean the fan blades, center hub, and surrounding cage. (Shroud)
Q-tips comes in handy here also.
You will find that after using the can of air for a while, that the air pressure drops.
This is because the air is coming out so fast, that the nozzle freezes.
When this happens set the can of air down for a while, and let the can warm up.
You will also find that if you tilt the can, some of the propellant shoots out. Try to refrain from tilting the can, but if you are doing a good job, this is inevitable.
Allow the computer to sit for at least 10 minutes after you are done cleaning, then close the case, and plug it in.
(The propellant is Flammable with most cans of compressed air. DO NOT use around sparks, or open flames. You are letting the computer sit for 10 minutes, so that any propellant that may have come out, will dissipate)
I would also advise removing all Ram Memory modules, ('Sticks'), and cleaning the gold plated contact pins on the bottom.
Use a pencil eraser to clean. Clean each contact pin on both sides. Doesn't take long.
You don't have to rub the eraser on the contact pin very hard.
In fact you may not even see that you have accomplished anything. I assure you that you have.
Corrosion develops on those gold plated contact pins.
Doesn't take very much to create a bad contact surface.
Use air to remove the eraser dust. If you have air leftover from the can of air, (Doubtful), use it.
Otherwise air pressure from your mouth will be sufficient.
Replace the Ram Memory modules correctly, amnd make sure they are seated well.
If you need guidance, please post in a Comment.
You may find that the Thermal Paste has dried up in-between the top of the Processor's case, and the bottom of the Heatsink.
Perform the above procedures, and if this is to no avail, post, and I will outline how to clean, and properly apply fresh, new Thermal Paste. (Thermal Paste is also known as Thermal Grease)
Dell Support > Inspiron 580S Desktop Computer > Service Manual,http://support.dell.com/support/edocs/systems/insp580s/en/sm/index.htm
1) Computer Cover = How to remove the computer case cover
2) Before You Begin/Safety Instructions = Advice on Anti-Static Precautions also.
(CAUTION at the bottom)