Is there a way to make Windows 7 do this: "10 minutes from now, close all applications and shut down (or hibernate), no matter what input you get from the keyboard or the mouse"? It's OK if the user can revert the shutdown order during the 10 minute period, but once the shutdown begins there should be no turning back.
6 Answers
The shutdown command line command is what you're after, with /t 600 you'll get a 600 second (10 minute) delay other options define the shutdown process. In your case I suspect you'll want shutdown /t 600 /s.
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As Havok stated, you can just use the Shutdown switch from command prompt. See options below.

Run from cmd.exe or just put into a .bat script as below (and you could just add this .bat to Windows scheduler if you wanted to schedule) - remove the "pause" if you want it to run straight away:
@cd /
@cls
@echo #Shutdown 10min script#
@pause
@shutdown /t 600 /s
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There are a number of automatical shutdown programs on the net. You can use one of them; for example: Sweet Dreams.

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This can be done by following the simple steps below:
- Go to run (Win + r)
- Type
shutdown -s -t 600and hit enter
The computer will shut down in exactly 10 minutes (600 seconds).
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I made a script for myself to shutdown the computer after a time interval that I set. this is the code:
:shutdown_sequence
set /p TM="Enter a time to shut down:"
set /a TM1=TM*60
echo I will shutdown the computer in %TM% minutes and in seconds: %TM1%
shutdown -s -f -t %TM1%
pause
exit
goto:eof