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# The rectangular solid above is made up of eight cubes of the same size

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Math Expert
Joined: 02 Sep 2009
Posts: 43377
The rectangular solid above is made up of eight cubes of the same size [#permalink]

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04 Dec 2017, 02:37
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The rectangular solid above is made up of eight cubes of the same size, each of which has only one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

(A) 1/6
(B) 3/14
(C) 1/4
(D) 2/7
(E) 1/3

[Reveal] Spoiler:
Attachment:

2017-12-04_1434.png [ 3.59 KiB | Viewed 429 times ]
[Reveal] Spoiler: OA

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Re: The rectangular solid above is made up of eight cubes of the same size [#permalink]

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04 Dec 2017, 10:10
Bunuel wrote:

The rectangular solid above is made up of eight cubes of the same size, each of which has only one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

(A) 1/6
(B) 3/14
(C) 1/4
(D) 2/7
(E) 1/3

[Reveal] Spoiler:
Attachment:
The attachment 2017-12-04_1434.png is no longer available

Attachment:

8cubes.png [ 5.47 KiB | Viewed 222 times ]

All eight equally sized cubes could have the one blue-painted side facing outward, see diagram, for example, facing us.

At most 6 sides of a cube are covered.
On the bottom and from the left, the second cube (among the most restricted), could have its blue side be on the surface in two ways: facing us, and on the bottom.
At the least, a cube has two "chances" of its painted side's being on the surface.
Hence all eight blue faces could be part of the surface area.

Total surface area in terms of faces of cubes: 8 + 8 + 4 + 4 + 2 + 2 = 28

Total painted faces: 8

Fraction of Painted area/Total S.A. =
$$\frac{8}{28}=\frac{2}{7}$$

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Re: The rectangular solid above is made up of eight cubes of the same size [#permalink]

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11 Jan 2018, 13:58
Bunuel wrote:

The rectangular solid above is made up of eight cubes of the same size, each of which has only one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

(A) 1/6
(B) 3/14
(C) 1/4
(D) 2/7
(E) 1/3

[Reveal] Spoiler:
Attachment:
2017-12-04_1434.png

If we let the side of each cube = 1, then the total surface area of the solid is:

2 x (4 x 2) + 2 x (1 x 2) + 2 x (4 x 1) = 16 + 4 + 8 = 28

We can also simply count the exposed faces of the solid: 8 in front side + 8 on the back side + 4 on top + 4 on the bottom + 2 on the left side + 2 on the right side = 28.

Let’s assume the front face of the solid is painted, i.e., all 8 faces of the cubes that are shown on the front face of the solid are painted.

Thus the painted area is 8 x (1 x 1) = 8 and, hence, the greatest fraction of the total surfaces area that is painted blue is 8/28 = 2/7.

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Re: The rectangular solid above is made up of eight cubes of the same size   [#permalink] 11 Jan 2018, 13:58
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# The rectangular solid above is made up of eight cubes of the same size

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