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Given a wine glass with angle theta, height h, what diameter ball, when placed in the wine glass will displace the most volume?
Your choice.tony873004 said:Does the ball need to fit completely in the glass, or can the ball sit on the rim of the glass, with its bottom protruding down into the liquid, which would make me want to say "on the glass" rather than "in the glass"?
Amongst balls that fit in the glass (depending on just what you mean by that)... why do you think the largest one displaces the most volume?If the ball is not allowed to sit on the rim of the glass, and the glass is an upside down circular pyramid, and the ball is assumed to sink, then you just need to compute the largest ball that will fit in the glass.
Hurkyl said:Amongst balls that fit in the glass (depending on just what you mean by that)... why do you think the largest one displaces the most volume?
That's why I said it depends on what you meant by "in the glass".tony873004 said:ok, I was thinking that it would have to completely fit in the glass, slightly smaller than your 2nd image. I guess I wasn't thinking outside the box, er... wine glass.
tony873004 said:ok, I was thinking that it would have to completely fit in the glass, slightly smaller than your 2nd image. I guess I wasn't thinking outside the box, er... wine glass.
Has a pretty answer too, at least if I haven't flubbed it up.uart said:BTW. It's an interesting problem Jeff
Hurkyl said:Has a pretty answer too, at least if I haven't flubbed it up.
I concur. Try drawing the solution.uart said:BTW Hurkyl, just a quick numerical check to see if my ugly solution could be correct. What do you get if you take theta = pi/3 (that's the full angle at the base of the glass ok) and height h = 10cm ? I get the radius of 5cm gives maximum displacement for this numerical case. How does that compare with your solution?
It factors. And I didn't use the angle until the very end of the problem; the arithmetic is probably a lot simpler that way.uart said:The solution was ugly because the coefficients of the quadratic where messy trig functions of the angle.
Yes, I meant martini glass.martini - cocktail glass.
My dad, who was a civil engineer (he passed away back in 1990) mentioned this problem to me and my half brothers. I'm not sure of the original source, but my guess is that it was typical of the type of math puzzles that math and engineering students got involved with back in the 1940's.uart said:It's an interesting problem Jeff, where did this one come from?
I know the answer is a rational function in sin(q), so it has the right form. I solved it w.r.t. x (where (h - x) sin q = r), and I was too lazy to convert from x to r.Jeff Reid said:Thanks, I'll have to search around through my "archives" to see if I got the same or similar answer. Hurkyl, did you get a similar answer?
It will not.Jeff Reid said:One of my half brothers mentioned that a 2 dimensional solution should probably work, but I never got a follow up from him.
The maximum volume of a ball that can fit inside a wine glass is approximately 60% of the volume of the wine glass. This is because the ball must be able to fit through the opening of the wine glass and still have enough space to move around and settle.
The optimal shape for a ball to maximize its volume in a wine glass is a sphere. This is because a sphere has the smallest surface area for a given volume, allowing it to fit into the wine glass with the least amount of wasted space.
Yes, the material of the ball can affect its ability to maximize its volume in a wine glass. For example, a ball made of a compressible material, like foam, can be compressed to fit through the opening of the wine glass and then expand to fill the space inside, allowing for a larger volume.
Other factors that can affect the volume of a ball in a wine glass include the size and shape of the opening of the wine glass, the thickness of the glass, and the amount of liquid already in the glass. These factors can impact the available space for the ball to fit and move around.
Yes, there are practical applications for maximizing the volume of a ball in a wine glass, such as in the design of packaging or containers for fragile items. Knowing the maximum volume of a ball that can fit inside a certain size and shape of a container can help in creating efficient and protective packaging solutions.