One advantage is that thinner air results in less drag, which also results in less fuel consumption. That’s pretty straightforward.
Another advantage is that if something goes wrong, you have a lot more time to react to it at high altitude. Even if you have a total engine failure you can still glide for a long time before you crash, and the higher up you are, the longer and further you can glide. That gives you time to possibly restart the engines, or glide down to a runway somewhere nearby and land safely.
One last advantage is that pretty much any long range radar can see you clearly at high altitude. So wherever you happen to be, there’s most likely at least one person on the ground who has you on their scope at all times. Even over the ocean there’s still probably a warship somewhere that has you on their scopes. So, if your navigation systems fail, you can always radio the people on the ground to avoid getting lost.
Another thing to add to that is that every commercial airliner has a flight plan that specifies how high they should be flying. So if your radio goes out, you can descend a few thousand feet, and the people watching on the ground will try to call you and figure out why you’ve deviated from the flight plan. When they can’t get ahold of you, they’ll send up a fighter jet to make visual contact and establish communication via hand signals.
Edit: one last thing i forgot. Flying that high also lets you fly above storms and other bad weather. If you’re above the clouds, you won’t have to deal with rain, hail, lightning, or overall poor visibility.
Edit 2: last one i promise. Birds also can’t fly that high. Bird strikes already do a lot of damage to aircraft as it is, so flying up that high eliminates the risk of hitting them.
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