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What is the volume of a certain cube?

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What is the volume of a certain cube? [#permalink] New post 10 Dec 2013, 13:12
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What is the volume of a certain cube?

(1) The sum of the areas of the faces of the cube is 54.
(2) The greatest possible distance between two points on the cube is \(3\sqrt{3}\)
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Re: What is the volume of a certain cube? [#permalink] New post 10 Dec 2013, 13:29
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What is the volume of a certain cube?

(1) The sum of the areas of the faces of the cube is 54.

Because this is a cube, all surface areas must a.) be equal to one another and b.) must have the same length and width.

I.) divide the sum of the surface areas into six equal parts (for the six faces of a cube)
54/6 = 9
area (of one face of cube) = l^2 --> 9 = L^2 --> L=3 Volume = l^3 --> v=27.

(2) The greatest possible distance between two points on the cube is 3\sqrt{3}

The greatest possible distance from one point to another in a cube is from one angle down to it's opposite angle (as shown in the diagram with a purple line) As you can see, the purple diagonal forms a right triangle inside the square. Therefore, the greatest possible distance can be found by finding the hypotenuse of the right triangle using the Pythagorean theorem:

a^2 + b^2 = c^2 --> x^2 + (x√2)^2 = (3/√3)^2
x^2 + x^2/2 = 3
2x^2/2 + x^2/2 = 3
3x^2/2 = 3
x = √2

So the side length of this cube is √2. Therefore, the volume of the cube = √2^3. Sufficient.

D
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Re: What is the volume of a certain cube? [#permalink] New post 10 Dec 2013, 13:54
Dear WholeLottaLove,

In statement two, I calcilate X=1. Can you explain how you get X=√2.

Thanks in advance.

Papua
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Re: What is the volume of a certain cube? [#permalink] New post 11 Dec 2013, 01:58
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papua wrote:
Dear WholeLottaLove,

In statement two, I calcilate X=1. Can you explain how you get X=√2.

Thanks in advance.

Papua


Actually x should be 3, there.

What is the volume of a certain cube?

(1) The sum of the areas of the faces of the cube is 54. A cube has 6 faces and the area of each is a^2, where a is the length of a side. Thus we have that \(6a^2=54\) --> \(a=3\) --> \(volume=a^3=27\). Sufficient.

(2) The greatest possible distance between two points on the cube is \(3\sqrt{3}\). The implies that the diagonal of the cube is \(3\sqrt{3}\) --> \(diagonal=a^2+a^2+a^2=(3\sqrt{3})^2\) --> \(a=3\) --> \(volume=a^3=27\). Sufficient.

Answer: D.
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Re: What is the volume of a certain cube? [#permalink] New post 11 Dec 2013, 02:03
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Bunuel wrote:
papua wrote:
Dear WholeLottaLove,

In statement two, I calcilate X=1. Can you explain how you get X=√2.

Thanks in advance.

Papua


Actually x should be 3, there.

What is the volume of a certain cube?

(1) The sum of the areas of the faces of the cube is 54. A cube has 6 faces and the area of each is a^2, where a is the length of a side. Thus we have that \(6a^2=54\) --> \(a=3\) --> \(volume=a^3=27\). Sufficient.

(2) The greatest possible distance between two points on the cube is \(3\sqrt{3}\). The implies that the diagonal of the cube is \(3\sqrt{3}\) --> \(diagonal=a^2+a^2+a^2=(3\sqrt{3})^2\) --> \(a=3\) --> \(volume=a^3=27\). Sufficient.

Answer: D.


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Re: What is the volume of a certain cube? [#permalink] New post 11 Dec 2013, 02:05
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WholeLottaLove wrote:
What is the volume of a certain cube?

(1) The sum of the areas of the faces of the cube is 54.

Because this is a cube, all surface areas must a.) be equal to one another and b.) must have the same length and width.

I.) divide the sum of the surface areas into six equal parts (for the six faces of a cube)
54/6 = 9
area (of one face of cube) = l^2 --> 9 = L^2 --> L=3 Volume = l^3 --> v=27.

(2) The greatest possible distance between two points on the cube is 3\sqrt{3}

The greatest possible distance from one point to another in a cube is from one angle down to it's opposite angle (as shown in the diagram with a purple line) As you can see, the purple diagonal forms a right triangle inside the square. Therefore, the greatest possible distance can be found by finding the hypotenuse of the right triangle using the Pythagorean theorem:

a^2 + b^2 = c^2 --> x^2 + (x√2)^2 = (3/√3)^2
x^2 + x^2/2 = 3
2x^2/2 + x^2/2 = 3
3x^2/2 = 3
x = √2

So the side length of this cube is √2. Therefore, the volume of the cube = √2^3. Sufficient.

D


You should have spotted that there was something wrong with your solution since you got two different values for the volume from (1) and (2). On the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other or the stem.
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COLLECTION OF QUESTIONS:
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Re: What is the volume of a certain cube? [#permalink] New post 11 Dec 2013, 13:04
Thanks Bunuel for your explanation.
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Re: What is the volume of a certain cube? [#permalink] New post 12 Dec 2014, 06:05
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Re: What is the volume of a certain cube?   [#permalink] 12 Dec 2014, 06:05
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