2009-01-22

More Solid Geometry and Booze

Bengt is having a birthday party and wants to offer his guests something really special. He discusses with Sture what kind of drink you could give the guests to impress everyone.
"It has to be something that no one has experienced before." says Bengt. "Were you at Percy's party last week?"
"No, what drink did he have?" asks Sture.
"A 'Bloody Mormon'; one part vodka, one part juniper berry and one part meerkat blood."
"That's not very imaginative. I had one of those the already in 1963."
"But hey, Gilbert had a really interesting drink at his party last year. A saturated solution of salmiac (NH4Cl) in water, served in half-spherical glasses with radius 5 cm."
"Yeah, it had quite a bite. That won't be easy to beat."
"Ah, don't worry. You don't play in the top league unless you're good. I think I have an idea, actually. What are the two most important properties of a good drink?"
"Strong and cold."
"Exactly. So how does this sound: More than a hundred percent alcohol, and a serving temperature of minus 104 degrees?
"That sounds absolutely marvellous."
"I was planning to use some rather special cups to measure the drinks. I have ten canisters, shaped as each of the platonic bodies, five with the side one dm and five with the side √10 dm. The thickness of the walls of the canisters is negligible. So you use them to measure the drinks, and then you pour them in big glasses."
"And then you pour them in your own non-platonic body."
"Exactly."
  1. Bengt would like to be able to measure 1, 1/2, 1/3, 1/4 and 1/5 liters, using only completely filled canisters. Is it possible? Which ones are possible?
  2. Unfortunately Gilbert had a marble floor at his party. If anyone spilled a drink, what volume of the floor was dissolved? (Assume that the glass was completely filled.)
  3. Bengt would like to have very long straws to the drinks. How long straws is it theoretically possible to have? (Assume that the drink has the same density as water, and that the drink is on Earth, and that the straws are pointing straight up.)
  4. Actually, the density of this particular drink is 568 kg/m^3. So how long can the straws be?
  5. The drink which Bengt has been planning consists of liquid ethene, which reacts with water in the stomach to form ethanol. What volume percentage of ethanol does this drink correspond to?