The figure above represents a box that has the shape of a cube. What i

Hi All,

We're told that the figure above represents a box that has the shape of a CUBE. We're asked for the is the volume of the cube. While this question might appear a bit 'scary', there's a great 'logic shortcut' built into it - since we're dealing with a CUBE, we know that all of the dimensions are EQUAL. By extension, if we know ANY length connecting two of the 8 vertices on the cube, then we can figure out ALL of the other lengths (using other Geometry formulas, although we won't actually have to do any of that math here) - and ultimately determine the volume.

1) PR = 10 cm

Length PR is a diagonal that forms on each face of the cube, so it would be the hypotenuse of a 45/45/90 right triangle. With that measurement, we could calculate the exact values of the sides and calculate the volume. There would be only one answer.

Fact 1 is SUFFICIENT

2) QT = 5√6 cm

When dealing with a 'rectangular solid', the formula for calculating the length from one 'corner' of the shape to the 'opposite opposite' corner is:
√(L^2 + W^2 + H^2)

Since we're dealing with a cube, we know that the length, width and height are the SAME. We can refer to all of those lengths as "X", which gives us:
√(X^2 + X^2 + X^2) = √(3X^2) = 5√6

With one variable and one equation, we CAN solve for the value of X - and there would be just one value, so we could calculate the volume of the cube.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Solution:

Question Stem Analysis:

We need to determine the volume of the box shown. Notice that if we can determine the side length of the cube, then we can determine the volume of the cube.

Statement One Alone:

Since we are given the length of a face diagonal of the cube (i.e. the diagonal of a face of the cube), we can determine the side length of the cube and hence the volume of the cube. Statement one alone is sufficient.

Statement Two Alone:

Since we are given the length of a space diagonal of the cube (i.e. the diagonal that passes through the center of the cube), we can determine the side length of the cube and hence the volume of the cube. Statement two alone is sufficient.

(Note: either statement will yield 5√2 as a side length of the cube and hence the volume of the cube is (5√2)^3 = 250√2 cm^3.)

Answer: D

The Logical approach to this question is simple and extremely fast: when it comes to regular solids - just like regular polygons - any given measurement (side, diagonal, surface area, volume, etc.) is enough on order to find any other measurement. Thus, since both statements provide such measurements, they each suffice on their own and the correct answer is (D).

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could you kindly elaborate on the equation more please?

We just need to know one side value since volume = s^3

(1) we are told PR (s root 2 = 10) --> we can determine S from here
Sufficient

(2) Diagonal of a cube = s root 3
s root 3 = 5 root 6
We can find s from here
sufficient

D

Hi All,

I was stumped by Option 2 of this question.
I have gone through the OG2020 Math section for concepts. I could not find the formula for the diagonal of a cube in it. Which material would you suggest I should study further for such concepts?

Thank you in advance.

Hi jimmyvithalani,

The formula for the length of a diagonal that runs through a rectangular solid (re: a cube or other "uniform box") is: √(L^2 + W^2 + H^2)

This is an incredibly rare issue on the GMAT - and it's likely that you will NOT face any questions on Test Day that will require that you have this knowledge. As it stands, you do NOT need to know this formula to answer the given question IF you recognize that you can form a right triangle on the "floor" of the box and another right triangle "standing up inside the box." The rarity of this type of question is probably why you would not have been taught the concept as part of your studies. By extension, if you are not close to your Score Goal yet, then these types of rare Quant and Verbal rules are NOT what you should be focused on. There are a number of other big-point categories that are far more important.

1) How have you been scoring on your practice CATs/mocks?
2) What is your overall Goal Score?

GMAT assassins aren't born, they're made,
Rich

Hi Rich,

Thank you for your response.

I could visualize the two triangles you mentioned and could come up with the formula that was used.
I also understand your point about focusing on the other more common concepts first. I am around 50 away from my goal score, so it makes sense to focus on some of the big-point areas like you mentioned.

Cheers,
Jimmy.

Hi jimmyvithalani,

Rather than continue this conversation in this thread, if you're interested in discussing your studies in more detail (and what you should be focusing on at this point), then you can feel free to PM me directly. I've sent you a PM with some additional questions about your studies and goals.

GMAT assassins aren't born, they're made,
Rich

The volume of a Cube \(= S^3\)

(1) Side of a cube \(= \frac{D}{\sqrt{2}}\) For the smaller digoanl; Sufficient.

(2) Side of a cube \(=\frac{D}{\sqrt{3}}\); For the larger digoanl; Sufficient.

The answer is D

Answer: Option D

Video solution by GMATinsight

[/quote]

The volume of a Cube \(= S^3\)

(1) Side of a cube \(= \frac{D}{\sqrt{2}}\) For the smaller digoanl; Sufficient.

(2) Side of a cube \(=\frac{D}{\sqrt{3}}\); For the larger digoanl; Sufficient.

The answer is D[/quote]

Where did you get these formulas to calculate the side of a cube from the face diagonal and the longest diagonal?? Haven't seen them anywhere before. The face diagonal I can understand - it comes from the 45-45-90 triangle, but where did you get the other one?

Video solution from Quant Reasoning:
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