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Two objects, Three views, Identical shapes


Best Answer harey, 18 December 2013 - 04:56 PM

Spoiler for You cannot tell
Go to the full post


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#1 bonanova

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Posted 17 December 2013 - 07:38 PM

The front, side and top views of two objects give identical appearances.
Object A appears in each case to be a square of side a.
Object B appears in each case to be a circle of diameter a.
If they are made of the same material, can you say which object is heavier?

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#2 BMAD

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Posted 18 December 2013 - 12:34 AM

Spoiler for

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#3 bonanova

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Posted 18 December 2013 - 12:03 PM

Spoiler for


Spoiler for say that

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#4 harey

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Posted 18 December 2013 - 04:56 PM   Best Answer

Spoiler for You cannot tell

Edited by harey, 18 December 2013 - 04:58 PM.

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#5 bonanova

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Posted 19 December 2013 - 05:22 PM

By same material I implied same density. Suppose d=1 so volume = weight.

Going a bit further just for fun, which object can weigh the most, and which can weight the least?
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#6 harey

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Posted 20 December 2013 - 12:40 PM

The circle is inscribed to the square...

 

Going a bit further just for fun, which object can weigh the most, and which can weight the least?

That's easy. The circle is inscribed to the square.


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#7 bonanova

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Posted 20 December 2013 - 02:55 PM

What if they were made of a thin film of small thickness d?
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#8 harey

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Posted 20 December 2013 - 08:54 PM

The best way is to make them of equal volume ( suppose what I call a "3D square" is a prism?):

 

v_prism=a*a*d_prism

v_cylinder=pi*a/2*a/2*d_cylinder

 

=> d_cylinder=(4/pi)*d_prism

 

While it might work for small d, with growing d, it will became more and more perceptible that the cylinder should be a kind of barrel (or a pancake). The formulae get too complicated for my taste and my possibilities.


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#9 bonanova

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Posted 20 December 2013 - 10:44 PM

I thought of the lightest "cube" as three squares joined at a corner, Volume = 3a2d.
The lightest "sphere" could be a "soap bubble'" Volume = pi a2d.
In that sense, the "cube" could be the lighter of the two.

Just a thought.

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#10 harey

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Posted 21 December 2013 - 06:03 PM

Yes, I see. There must be a lot of solutions, like taking a sphere and a cube and drilling holes into the cube until it becames lighter than the sphere - no formula, no misscalculation.

 

My preferred solution is still that of the three squares and three pankaces in the planes xy, xz and yz.

 

BTW, the Volume=3a2d-d3 - I have the same oversight in my post n. 8.

 

I just wonder how often we make false deductions in the real life in the style "three projections=circle => it must be a sphere".


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