What is the volume of a certain cube?

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.

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