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#1 2013-07-09 10:16:23

cooljackiec
Member
Registered: 2012-12-13
Posts: 160

rectangular prisms

Let  ABCDEFGH be a rectangular prism, as shown. The areas of faces ABCD, ABFE, and ADHE  are 56, 110, and 385, respectively. Find the volume of the rectangular prism.

The total area of all the faces of a rectangular prism is 22, and the total length of all its edges is 24. Find the length of the internal diagonal of the prism.

View Image: ScreenHunter_245 Jul. 09 15.15.jpg View Image: ScreenHunter_246 Jul. 09 15.16.jpg

I see you have graph paper.
You must be plotting something
lol

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#2 2013-07-09 11:19:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: rectangular prisms

Hi cooljackiec;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2013-07-09 19:28:44

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,394

Re: rectangular prisms

hi cooljackiec,

(i) The tetrahedron.  I think the little one has sides that are half the large one.  So that should fix the ratio of volumes.

LATER EDIT:  I've changed my mind about this one.  See post 10 on

http://www.mathisfunforum.com/viewtopic … 60#p277760

(ii) If you call the sides a, b and c then

ab = 56      ........................(p)
bc = 110     ........................(q)
ca = 385    .........................(r)

If you do equation (p) divide by (q) you'll eliminate b

Then make a the subject and substitute into (r) to solve for c.

Then you can substitute back to get a and b.

(iii) Again call the sides a, b and c.

From the information given

This time you cannot solve for a, b and c because there are only two equations.  But you don't need to, because we only want to get

There's a neat way to get this.  Start with

If you substitute in the values you know you'll be left with the value of a^2 + b^2 + c^2  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2013-07-10 06:49:45

cooljackiec
Member
Registered: 2012-12-13
Posts: 160

Re: rectangular prisms

hmmm... i used some weird guessing/intuition and got half of the correct answer of number 1...


I see you have graph paper.
You must be plotting something
lol

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#5 2013-07-10 08:40:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: rectangular prisms

Hi cooljackiec;

cooljackiec wrote:

hmmm... i used some weird guessing/intuition and got half of the correct answer of number 1...

That is how I get all my answers. Did you check the answer?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#6 2013-07-10 10:02:22

cooljackiec
Member
Registered: 2012-12-13
Posts: 160

Re: rectangular prisms

yes, yours was correct


I see you have graph paper.
You must be plotting something
lol

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#7 2013-07-10 10:08:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,705

Re: rectangular prisms

Over at Brilliant - AOPS they say "Work smarter, not harder." But where does that leave guys like me?

Arthur Ashe wrote:

Paralysis by analysis.

I say, take thinking out of the solution, it only gets you into trouble. Do not follow Karl Maldens advice, I say leave your brain at home, travel light!

Turn everything into a small template of computational problems and then work on those. The true "Teakettle Principle!"


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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