You were specifically told to Remember the Alamosaurus.
“Need answer fast!”
Just want to make a quick comment here, if we are talking about a military flamethrower (designed for combat) being “relatively short ranged”, we are talking about hundreds of yards. That’s not exactly knife-fighting range (unlike the propane flamethrowers you see in the movies, actual flamethrowers were kind of like supersoakers filled with jellied diesel fuel). The biggest problem would be lugging around the big backpack with the fuel and propellant tanks. Probably better off just carrying a BFG and lots of ammo.
Isn’t it more likely that the sound of any gun (maybe not a 22) would scare off the T. Rex?
I’m sure they’ve heard thunder. I wouldn’t count on scaring one away with a loud noise.
I wouldn’t want to count on it but a gun really doesn’t sound like thunder. I’ve been hunting and missed my target plenty of times. They always run away after hearing the shot. They don’t seem to think it’s thunder.
I’d give the T. Rex a BB gun. He’ll shoot his eye out with it.
(see also post #35)
George Orwell’s essay Shooting an Elephant gives a nice account of what happens when you don’t actually know where an animal’s brain is.
What are you hunting? I’d expect that from prey animals like deer, less so from cougars or other apex predators. Besides, modern animals may associate gunshots with humans and be understandably cautious. A T. Rex in some jungle nowadays might be the same way; otherwise, it wouldn’t have lasted so long without being found. A Cretaceous tyrannosaur might not.
There’s a theory that some sauropods may have been able to create sonic booms by whipping their tails. Maybe gunfire would sound like “Lunch Over Here” to a T-Rex! 
**Killing T-Rex with a gun **
That’s a ridiculous question. Where would a T-Rex get a gun? And how would he use it, lacking opposable thumbs?
Simple, just give him gatling gun arms. 
There is the theory that T. rex may have been a scavenger anyway. I’m not sure what the status of that idea is these days, but as far as I know it’s still considered a possibility. Also consider that humans are quite capable of scaring off bears or sharks.
A shot from 2-3 miles away? Good luck!
While the ‘effective range’ is only rated at 2,000 meters or about 1.25 miles, the maximum range is over 4 miles. In this case ‘effective’ means able to penetrate body armor. While a 2 mile shot wouldn’t reach the t-rex with maximum energy it would definitely sting. Being able to continuously fire at him for up to 10 minutes I think it would do the job and after half the time he would have moved into ‘effective range’ (if he could even discern where the threat was originating).
Yeah, but you’re got to aim it too. That effective range is a bit less than the record for the longest sniper kill and there are only 4 others recorded at over 2,000 metres. Granted, a T. rex is a much bigger target, but at that kind of range you have to take into account bullet drop, wind and the Coriolis effect, as well as holding the rifle very steady. And then at that range you would probably not know where the bullet did go after it missed, making it harder to correct. I’m not sure to exactly what degree these factors affect a bullet’s trajectory though, so perhaps a half-decent shot would still hit it somewhere despite them. Of course if the T. rex isn’t coming after you (it seems like cheating if it’s not but that’s not in the “rules”) it would be a lot easier, as you said.
I like the multiple shots at the T-rex from a long range with a .50 cal sniper rifle strategy as the T-rex approaches strategy. Reminds me of the game OGRE. By analogy with OGRE, I’d aim for the legs just like those OGRE games you won by taking out the treads using mostly artillery.
Of course, I agree with the above posters who have said that predators are risk averse. I think the first time you stung it with the rifle it would prance around in different directions until it stopped getting stung. Might not even be smart or well informed enough to associate the sound of gunfire with the sting. I also agree that a flamethrower is a good choice. I am pretty sure a T-rex will run from fire.
But all that presumes the goal is to avoid getting killed by a T-rex, not that we were trying to kill one. Given odds are good that it will retreat from something totally novel, don’t we need a vehicle to pursue it? I think one of those old anti-aircraft vehicles with the quad-.50 machine guns would do the trick. The M2 Bradley vehicle would be even better. That 25mm chain gun has gotta hurt, and the TOW missiles have gotta be a one shot kill. The only question is how fast can the T-rex run? I know some might think it unfair to bring vehicles into this, but if we’re gonna kill it, we need to be able to chase it.
Of course the OP specified killing a rex that was attacking you, not hunting one from outside it’s detection range, just sayin’
T-Rex is famously allergic to peanuts as discovered by Jif’s time travel division of choosy moms.
A .50 caliber machine gun mounted on jeep like The Rat Patrol (in color).