What is the volume of a certain cube?

Question

(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}

WholeLottaLove · Answer

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

papua · Answer

Dear WholeLottaLove,

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

Thanks in advance.

Papua

Bunuel · Answer

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.

Bunuel · Answer

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Bunuel · Answer

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.

papua · Answer

Thanks Bunuel for your explanation.

Archit3110 · Answer

#1
area of cube = x^2
and 6 faces of cube ; 6x^2=54 x=3
sufficient
#2
s√3=3√3
s=3
sufficient
IMO D

IanStewart · Answer

For this kind of question, it can be instantly clear, with no calculation, that the answer must be D, because only one possible size of cube could satisfy each Statement. The larger you make a cube, the larger the area of its faces will be, and the larger the maximum distance between two points will be. There's no way you could have two different solutions (in other words, two different-sized cubes) that 'work' using either statement alone.

DanTheGMATMan · Answer

1) 6 s^2  = 54
 Solve for s

2) The longest distance between any 2 points of a cube is just the diagonal, which is side \sqrt{3}

Solve for s again.

D

What is the volume of a certain cube?

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What is the volume of a certain cube?

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