2008-06-24

The Alumium Can

"Consider this can." says Bengt.
"Okay." says Sture.
"It is a can of beer."
"Ah. One of the great inventions of mankind."
"Indeed. However, I'm thinking of returning this particular can to the store. It's flawed."
"How?"
"Well, we can assume that the outside shape of the can is a cylinder."
"Indeed."
"The space inside it is also a cylinder. The can itself is thus a cylinder minus another cylinder."
"Indeed."
"So the top and bottom are also cylinders, and they have the same thickness. The side wall has the same thickness all around, but not the same as the top and bottom. You'd expect all the walls to be very thin, but they're not; it's a very sturdy can."
"Right."
"Inside the can is beer and air. The beer has roughly the same density as water, but the difference should not be neglected. The density of the air, however, can be neglected. The can itself is made of pure alumium."
"You mean aluminium? Or aluminum, as the Americans say?"
"Well yes, but it should properly be called alumium."
"I see. So what's the problem with it?"
"It's not full. I don't know what fraction of it is full, and I have to find out without opening the can, because otherwise I obviously can't return it."
"Oh. Got yourself a good problem there, Bengt."
"Indeed."

Let the thickness of the top and bottom walls be b, the thickness of the sides be s, the density of the beer d, and the fraction of the inside which is filled with beer f.

  1. If you know b, s and f, you can find d by weighing the can. Obviously you can also measure the outside size of the can. What is the formula for d?

  2. If you know d, how can you find b, s and f? You must not open the can, or deform it. Bengt also does not have an x-ray camera. It's a mechanics problem, folks!

  3. If you don't know either of those variables, how would you go about solving it? Is it still possible?