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# A three-dimensional "skeleton" rectangular shape

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A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 09:45
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A three-dimensional "skeleton" rectangular shape
[Reveal] Spoiler: OA

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Veritas_Geometry.jpg [ 26.95 KiB | Viewed 1090 times ]

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Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 09:52
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Expert's post
Dear Rajathpanta,

I'm happy to help.

The skeleton is made of 12 rods --- -four forming the square base, four tall ones providing the height, and four forming the square top.

The square base has four rods, each of length x, so the total rod length of the base is 4x. Of course, the square at the top is identical, so this also has 4x of rod length, and together, they have 8x of rod length.

We don't know the height --- call it h. We have four rods of length h --- that's 4h.

When we add all twelve rod lengths together, that's 8x + 4h, and this has to equal the total, 480.

8x + 4h = 480
divide both sides by 4 to simplify

2x + h = 120

h = 120 - 2x

Does all this make sense?

Mike
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Mike McGarry
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Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 10:38
mikemcgarry wrote:
Dear Rajathpanta,

I'm happy to help.

The skeleton is made of 12 rods --- -four forming the square base, four tall ones providing the height, and four forming the square top.

The square base has four rods, each of length x, so the total rod length of the base is 4x. Of course, the square at the top is identical, so this also has 4x of rod length, and together, they have 8x of rod length.

We don't know the height --- call it h. We have four rods of length h --- that's 4h.

When we add all twelve rod lengths together, that's 8x + 4h, and this has to equal the total, 480.

8x + 4h = 480
divide both sides by 4 to simplify

2x + h = 120

h = 120 - 2x

Does all this make sense?

Mike

Hi Mike,

Unable to visualize... (highlighted)
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Kudos drives a person to better himself every single time. So Pls give it generously
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Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 10:55
Since base is a square 8x would be the total length of the two squares in the figure.
This leaves 4h (4 x number of rods representing the height) to complete the total 480 length of the figure.

Thus... 8x + 4h = 480.
This reduces to h = 120-2x
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Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 11:58
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Expert's post
rajathpanta wrote:
Hi Mike, Unable to visualize...

OK, it's a bit challenging to explain visualization via internet text, but here goes.

For this problem, it doesn't matter at all whether we are dealing with the skeleton frame or a solid box --- the math is exactly the same either way, so I am going to discuss a box, because that's easier to visualize.

Imagine we have a box sitting on a table --- say a carton around the size of a half-gallon of milk or orange juice, but flat at the top instead of peaked. This box has 12 edges. Any rectangular solid in the world has 12 edges. I strongly recommend you get a physical box of anything, as long as its fully rectangular, and trace your finger along the 12 edges to convince yourself of this fact. ----- An "edge" is the "corner line" where two flat faces meet --- any rectangular solid has six faces, the flat surfaces where they print stuff, and twelve edges, the lines where two faces meet, and six vertices, the corners where three edges come together.

Now, back to our box sitting on the table. First, we are going to take account of the four edges that are touching the table, the four on the bottom, surround the square on the bottom. There are four lengths of x, for a grand total of 4x for the lengths on the bottom.
Attachment:

box top & bottom.JPG [ 27.01 KiB | Viewed 1044 times ]

Now, consider the top of the box, the four edges that surround the square on the top, which is congruent to the bottom and parallel to it. This also has four lengths of x, so this also has a grand total length of 4x. If we add the lengths of the bottom and the lengths of the top together, that is 4x + 4x = 8x of wire needed to surround those two faces.

Now, we have to consider the four remaining edges. These edges have different lengths from the other eight. Unlike the bottom, which sits on the table, and the top, which is parallel to the table, these last four edges are perpendicular to the surface of the table. The make a right angle with the surface of the table and move vertically away from the plane of the table. These each have a length of the height --- the height is unknown, so I will just call it "h."
Attachment:

box sides.JPG [ 16.45 KiB | Viewed 1045 times ]

There are four of these edges, each with length h, so that's a total length of 4h for these final four edges.

If we add all twelve edge lengths, we get 8x + 4h = 480, which, as I explained above, leads to h = 120 - 2x.

Does this make more sense now?

Mike
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Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 12:07
Rectangular 3 Dimensional shape (parallelepiped) will have 6 surfaces & 12 edges. In this case out of 6, two surfaces are squares with side X and remaining 4 are rectangles with height H and width X.

Consider this rod (length=480) is split to form the 12 "edges" of this 3D rectangular shape.

Sum(length of all edges of shape) = sum (length of 4 edges of bottom square) + sum (length of 4 edges of upper square) + sum (length of 4 edges as heights)

$$480 = 4X + 4X + 4H$$

$$4H=480-8X$$

$$H=120-2X$$

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Senior Manager
Status: Prevent and prepare. Not repent and repair!!
Joined: 13 Feb 2010
Posts: 275
Location: India
Concentration: Technology, General Management
GPA: 3.75
WE: Sales (Telecommunications)
Followers: 9

Kudos [?]: 57 [0], given: 282

Re: A three-dimensional "skeleton" rectangular shape [#permalink]  15 Jan 2013, 20:17
I completely misread!!! thanks so much !! this was pretty easy....
_________________

I've failed over and over and over again in my life and that is why I succeed--Michael Jordan
Kudos drives a person to better himself every single time. So Pls give it generously
Wont give up till i hit a 700+

Re: A three-dimensional "skeleton" rectangular shape   [#permalink] 15 Jan 2013, 20:17
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