Math Is Fun Forum

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

some 3d stuff :D

We are given a cube of side length 1. We then slice a pyramid off each corner, as shown, so that every side length of the remaining polyhedron has the same length. Let A, P, Q, and R be the vertices shown.

Cache.artofproblemsolving.Com/asyforum/8/1/a/81ad9bccc16be74ecf4a456247ceaf4290418ffc.png

import three;unitsize(1 cm);currentprojection=perspective(6,3,2);triple A, B, C, D, E, F, G, H;A = (1,1,0);B = (1,0,0);C = (0...

(a) Let x = AP, y = AQ, and z = AR. Prove that x = y = z.

(b) Find x.

(c) Find the volume of the remaining polyhedron.

wow this ones killing me

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

hi thedarktiger,

We know that each face has eight equal sides.  Now you can have an octagon with equal sides that does not have equal angles but not when you start with a square.  So  angle AQR = ARQ = 45.  Let each side be s.

=> AR = AQ = side.sin45  = s.√2/2 = 0.707s

Similarly, AP.

And 0.707s + 0.707s + s = 1   => s = 0.414

Now to get the volume of one cut off pyramid.

The base is equilateral with side s, and the sloping edges have length 0.707s

Divide the equilateral in half with a line, length s.√3/2.

The centre of the base is one third of the way up this line (or two thirds from the vertex) = s.√3/3

so the height, h, of a pyramid  is given by h^2 = (s.√2/2)^2 - (s√3/3)^2 = s^2/2 - s^2/3

=> h = s.√(1/6)

Then the volume of one is one third base area times height.

Subtract 8 one these from the volume of the cube and you're finished! 

Bob

Last edited by Bob (2014-02-15 20:47:28)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: some 3d stuff :D

Wow thanks!

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Good. You can read.

SolarDevil

Member

Registered: 2014-09-14

Posts: 37

Re: some 3d stuff :D

I don't get how you get x. Could you help?

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Herro! Sycamore School will win National Science Bowl this year!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

x = AP

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SolarDevil

Member

Registered: 2014-09-14

Posts: 37

Re: some 3d stuff :D

So x is $\frac{2-\sqrt{2}}{2}$

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Herro! Sycamore School will win National Science Bowl this year!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

Correct.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SolarDevil

Member

Registered: 2014-09-14

Posts: 37

Re: some 3d stuff :D

Hmmm... well. When I enter it in it says that x is incorrect. What is wrong?

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Herro! Sycamore School will win National Science Bowl this year!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

When I enter it in it says that x is incorrect

Where did you get this problem from?  I have not tried 'entering it' anywhere, but I will if you post the link.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SolarDevil

Member

Registered: 2014-09-14

Posts: 37

Re: some 3d stuff :D

Nevermind. I have got it solved

Solar

(btw this is aops geo stuff)

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Herro! Sycamore School will win National Science Bowl this year!!!

cuiziyong

Member

Registered: 2015-02-09

Posts: 1

Re: some 3d stuff :D

SolarDevil that answer was not correct though...What did you enter?

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: some 3d stuff :D

I tried the problem too and got the same answer solar posted. However it was wrong.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

hi Enshrouded_

It was a long time ago and I've forgotten all about this problem.  So I have re-worked it from scratch.  I still think that post 2 is correct.

And that still gives

The original link in post 1 isn't working for me.  Please provide a link to this problem so I can see it in its original form and try my answer.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: some 3d stuff :D

I've tried how you told us to do the problem, but I got the answer wrong too.

Here's the link:
http://latex.artofproblemsolving.com/b/ … 1f27d7.png

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

hi dazzle1230

Thanks for the picture.  That is the diagram I was working with.  But what is the exact wording of this puzzle? Maybe I am trying to solve a different question.

If you post your working I can try to see if one of us has made an error.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: some 3d stuff :D

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some 3d stuff :D

hi thickhead,

In the original post x = AP (= AQ = AR to be proved) and I used s for the side length.

I got

and

Do you think I've got these wrong?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: some 3d stuff :D

I am sorry. I thought x is side length. You are correct.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: some 3d stuff :D

I see how you've got your answer but the answer still isn't correct.  I don't understand why though.

Here's the exact wording of the problem:

We are given a cube of side length 2. We then slice a pyramid off each corner, as shown, so that every side length of the remaining polyhedron has the same length. Let A, P, Q, and R be the vertices shown.  Let x = AP, y = AQ, and z = AR. Convince yourself that x=y=z. Don't forget here that all we know is that the edges of all of the octagons are the same length.  Find x.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: some 3d stuff :D

Here lies the point . We worked on cube of side length 1.
accordingly

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}