I figured that a ball of cheese weighing one pound would be about 4 or 5 inches in diameter. Because volume is calculated using the third power, I figured that I only had to multiply my diameter by 1000 (because 1000^3 is equal to a billion, which is close to the 1.39 billion figure in the OP). To simplify my mental calculation, I settled on 4800 inches, because that is equal to exactly 400 feet. I voted for the closest choice, 330 feet.
Well that’s going to make a bit of a change, isn’t it? You’re absolutely right.
So, let’s shrink the radius by a factor of 10 (cube root of a 1000.) And make the diameter about a football field instead.
Is the national sliced almond stockpile adequate for a cheese ball of this magnitude?
OK without looking anything up. Go with a “pint is a pound the world around” 1.39 billion pounds of water would be 1.39 billion pints of water. There are ~ 60 pints in a cubic foot so about 20+ million cubic feet of water. The radius of a sphere is about 4r[sup]3[/sup]. So r[sup]3[/sup] = 5 million so r is about 200 feet meaning D is about 400 feet. Cheese is somewhat denser than water so a bit smaller.
I’ll go with 330 feet.
But, but… (from the OP)
Why are we stockpiling so much cheese? 
Are we in fear of some sort of “cheese gap”?
My wife went to Dirty Y Town to visit her mom, and said the new store there was fresh out!
I think it has something to do with government subsidies to dairy farmers to keep the prices of milk high, or something. That’s my total WAG. Too lazy to look things up. Also, when living in our mineshafts, who wouldn’t want a grilled cheese sandwich?
[Emphasis mine]
Wait a minute. I’m fairly math-illiterate, but this doesn’t seem right to me. Isn’t the volume of a sphere of diameter d a bit more than half the volume of a cube with side d? Where’s my calculator?
I’m anxious and willing to be shown wrong.
sphere 4/3 pi r^3
cube (2r)^3 = 8 r^3
So the ratio is pi/6 = 0.52
Since that’s approximately 1 for estimation purposes, imo you are both correct!
“Without doing any research or math…”
That wasn’t math it was pure arithmetic. 
Lots of us answering did simple calculations like that. It’s simply the way we think. Is it math to think the opposite way? A pound of cheese is a few inches in diameter, volume and mass scale with the cube of the diameter? I guess that’s math in a sense, but not really what I think of as math.
“It’s not much of a cheese shop, is it?”
And although it may have been mentioned up-thread, who do we have on deck to cut so much cheese? And just whose dog do we blame it on?
We are a Christian nation.
If it were 1000’, then the cheese would only be about a pound per cubic foot. Too low. If it were 100’, then the cheese would have to be a thousand pounds per cubic foot. Too much. 330’ sounds fine. As a double check, 330’ is about 100 m, or 1000 times 10 cm. Since 10 cm cubed is a liter, we have a billion liters, and therefore about a liter per pound. Or per kilo, whatever–the two are the same for Fermi purposes. So 330’ is close.
I assumed you were just supposed to give your gut intuition on how big it would be, without doing any calculations. But I agree you are far from the only one not to do so.
Since I foolishly read more of the thread before voting, I will not vote.
I’m confused. This is Lutheran cheese now?
I’ve seen five pound blocks of cheese so I know what they look like. I imagined a 10x10x10 pile of these bricks. That would be five thousand pounds of cheese and I visualized it as being around the size of a small car. So I pictured a 10x10x10 pile of those. That would be five million pounds of cheese and I visualized it as being around the size of a medium sized office building. I did one more expansion in my mind, picturing a 10x10 grid but only making it two high. This gave me an approximate visualization of what a billion pounds of cheese would look like. I felt 330 feet was the closest measurement of what I was picturing so that’s what I voted for.
eta: I know the OP said no math. But I think imaging ten things laid out in a row doesn’t count as math.
I’ll buy that, Sam. Works for me.
I was thinking the same thing. My church is an ELCA Lutheran church. I wonder what ELCA is, in that context? Paging kopek…
Thanks everyone. My answer is:
[spoiler]From the poll selections, the closest answer is 330 feet. The “right” answer is about 359 feet. For what it’s worth, I used Vox’s cheese density information for Land o’ Lakes Medium Cheddar Cheese, Yellow.
That block is 14.9" x 11.9" x 7.3" and weighs 43 lbs. So, it’s 43 lbs/0.749 cubic feet, or 57.41 lbs / cubic foot (a bit less than the density of water, about 62.4 lbs / cubic foot). Thus, a 1.39 billion pound cheese stockpile is 24.21 million cubic feet, which translates to a sphere of 358.9 feet.[/spoiler]The answer is a tick smaller than I thought it would be. That seems to be the most common wrong answer. Few people guessed smaller, which is what I expected. I’m surprised at how big some of the guesses were and that the distribution of guesses is not very normal.
And with colony collapse disorder, we’ll never have enough honey.
I hoped people would just give me their instinctual response but I asked people to “estimate,” which implies some arithmetic. I should have asked people to just guess.
My own approximation was something like yours. How big a pile would I need to give everyone in America a four-pound block of cheese? The problem is that I can’t visualize 300m+ Americans, let alone what they look like holding cheese blocks.
I can’t think without using math. I interpreted the “no math” restriction to mean no calculator or pencil-and-paper calculations.