I have asked many strangers "How much does a cubic mile of air at sea level weigh (1 x 1 x 1)? The typical answer has been “Nothing. Air is weightless.”
If that were the case aircraft could not fly, birds could not fly, insects could not fly. All of them counter their weight by pushing down slightly more than their own weight, in air.
So let’s see your answers and calculations. You won’t BELIEVE the incredible answer, really. (Standard pressure, viz. 14.7 PSI at sea level)
We have to do math to figure out the answer? Ah man.
Give me your best guess. I’ll pay you ten bucks if you’re within 25%.
I refuse to do calculations involving US customary units. SI is the only way to go.
We would need to know much more than has been provided, for example, the answer is going to depend heavily on the pressure, but the pressure depends heavily on the temperature; and you didn’t even address the temperature, so it is probably best to stop here.
PS: when you say weight, do you mean mass, or are you serious about asking for the weight? Weight is a vector quantity…
PS2: There is also going to be a problem since pressure is sensitive to altitude, but you told us to keep the altitude fixed for a box that is 1 mile tall! That is going to require a huge correction imo.
One can assume a location that make a “standard atmosphere” applicable, like U.S. Standard atmosphere (the table starting at page 50), freely downloadable from NASA. But it comes in tables, and one has to do 
loads of manual typing to extract the necessary values from the tables. It is probably available from somewhere in electronic form, but right now I’m too lazy to search for it.
Find the average mass density by means of integration. Which has to be done numerically through the above mentioned tables. But since you’re going to integrate anyway, it’s meaningless to do it this way. Better just integrate it and get the answer directly.
NO, one would NOT “need to know a lot more” than I provided.
Everything needed is there.
14.7 PSI is standard pressure at sea level. That’s what I said.
The answer is 5.6 million tons, an incredible answer which when I gave it to a 737 captain flying me back from Seattle, amused him to the extent that he invited me to be first to board the plane, before even First Class, and come to his cockpit for a photo op, in his seat, wearing his hat. Way more fun than any atheist could create and enjoy.\
This is (5,280 x 12)2 x (14.7psi – 11.907psi) = 11,262,469,453 pounds or 5.606 million tons
This thread is “Random Fun.” You people steal all the fun out of everything you touch with your bitterness and bad attitude.
You also need the temperature profile. Is it in the tropics or in the arctic? With or without temperature inversions? Dry or humid air?
NO, you do NOT need temperature if you have sea level pressure.
Humidity is irrelevant. 14.7 PSI. Stop trying to make yourself look intellectual by trying to prove me wrong. “Inversions.” Oh brother.
Just pathetic of you. Pathetic. You have destroyed the intent of “Random Fun.” THE POINT was the massive number.
You atheists drive everyone away who is not just like you - condescending, cock-sure, irrational, and full of one-liners to attack and prove your own superiority.
Those are all parameters a professional physicist would use when doing that sorts of calulations. Like I used to do a couple of decades ago, for work.
Edit: Massive number? If you integrate over the entire atmosphere, you get an even more “massive number”. And if you calculate the mass of the earth, you get an even more and biglier massive number. And if you estimate all the mass in the visible universe, an even whoppingly shitload more massive number. What’s your point?
If you want really big numbers, look at Graham’s number, and the other numbers that you can generate with Knuth’s up-arrow notation, or Busy beaver numbers, or any of the other algorithms that generate the REAL big-ass numbers.
Add to that density! A cubic mile of air is an inconsistent volume. The air 1-mile above sea level is not the same as the air at sea level. This question is just weird. At sea level, the air pressure is Denver, Colorado is about 1.6 kilometers (1 mile) above sea level. The atmospheric pressure of Denver drops to 0.85 kilograms per square centimeter (12 pounds per square inch) at this altitude. he typical pressure at sea level is 1013.25 millibars or 14.7 pounds per square inch.
AGREE!
An airplane wing has design called airfoils, that move the air on the top of the wing faster than the air below the wing, creating higher pressure under the wing.
This has to be a windup no?
As others have indicated, and even a dullard like me can Google in a few seconds.
"Air weighs 1.2g per cubic metre at 20deg Celsius and at sea level.
With different temperature or pressure, the density of air is:
d = p / (R x T)
d is the calculated density of air
p is the pressure
R is the specific gas constant (287.05)
T is the temperature.
With differing altitudes and any water vapour (humidity), the weight of air varies in a much more complex fashion."
Oh dear…this guy reminds of someone on the old Amazon chat rooms, who insisted a light year was a measurement if time.
The calculation is a very simple one. One multiplies the area of a square mile, times the differential between standard pressure, at sea level, and the pressure, given by tables, at, 1 mile above sea level.
The answer is delightfully huge. THAT is the point what went WHOOSH right over all of your tin-foil covered heads.
You drive sane people to insanity!
And what is the pressure 1 mile up? (you know, at the top of the box)? You see—as @Get_off_my_lawn hinted—we are going to need to know the pressure as a function of altitude.
You have pounds per square inch equal to pounds 
Also, just looking at the numerals, your answer to your own calculation is off; by like 4-5 orders of magnitude…something on order of 100,000 (units?)
Look up the pressure at 1 mile. Don’t be so lazy. Your aluminum, typically called “tin” foil hat needs some serious polishing.
No, I do not have psi = pounds. What I did was show the number of square inches times the pounds per square inch differential, which gives a product of … pounds. Really, you people try so hard to be right especially when you are wrong.
After we remove all the dimensionless stuff:
(5,280 x 12)2 x (14.7psi – 11.907psi) = 11,262,469,453 pounds
You clearly have pounds per square inch set equal to pounds. It is gibberish.
I take this to mean that you by now have realised that you need both pressure (p) and temperature (T), in order to calculate mass density (ρ) according to the ideal gas law, p = ρRT? Pressure can largely be assumed to decrease exponentially, but temperature depends on a lot of factors, hence, the mass density varies according to the same factors. That’s why researchers in the big and diverse field of atmospheric physics measure atmosphere temperature profiles regularly. Meteorologists do the same, because they depend on temperature profiles to make accurate weather forecast.
Edit: besides, US customary units are an abomination, at least for science and engineering purposes.
I actually love the fact that we have people who understand higher mathematics on the site. LOL Not that you can do anything with math to prove a God. These dweebs get away with it, because the people around them don’t understand what they are talking about. When they meet someone that actually understands the math they are trying to use, it just exposes their ignorance. BONUS POINTS AWARDED!