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555-605 (Medium)|   Geometry|      
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Bunuel
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The diagonal of one face of cube P is three times as long as the diago 09 Sep 2018, 07:58
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35% (medium)

Question Stats:

70% (01:28) correct 30% (02:06) wrong based on 37 sessions

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The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64
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D
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The diagonal of one face of cube P is three times as long as the diago 09 Sep 2018, 09:27
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Bunuel
The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64
Show more

Diagonal of a cube=\(\sqrt{3}\)*side length

Given, the diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q.
Or, \(\sqrt{3}\)*side length of cube P=3*\(\sqrt{3}\)*side length of cube Q
Or, side length of cube P=3*side length of cube Q

Volume of cube Q=\((1/3*Side-length-of-cube-P)^3\)=\(\frac{1}{27}\)* Volume of cube P

Ans. (D)
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The diagonal of one face of cube P is three times as long as the diago 11 Sep 2018, 06:30
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PKN
Bunuel
The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64

Diagonal of a cube=\(\sqrt{3}\)*side length

Given, the diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q.
Or, \(\sqrt{3}\)*side length of cube P=3*\(\sqrt{3}\)*side length of cube Q
Or, side length of cube P=3*side length of cube Q

Volume of cube Q=\((1/3*Side-length-of-cube-P)^3\)=\(\frac{1}{27}\)* Volume of cube P

Ans. (D)
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Why is the diagonal of a face of a cube \(\sqrt{3}\)*side length and not \(\sqrt{2}\)*side length according to our 45-45-90 right triangle?

It's the same answer in the end, though
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The diagonal of one face of cube P is three times as long as the diago 11 Sep 2018, 18:50
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Bunuel
The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64
Show more
Pick numbers. If the diagonal of one face of cube A is three times as long as the diagonal of cube B, the side of A will be three times the side of the B.

Let side \(A = 3\)
(diagonal would be \(3\sqrt{2}\))

Let side \(B = 1\)
(diagonal would be \(1\sqrt{2}\))

Volume of A = \(3^3=27\)
Volume of B = \(1^3=1\)

\(\frac{Vol_{A}}{vol_{B}}=\frac{27}{1}=27\)

Answer D
Scale factor
Two similar figures such as A and B above have corresponding sides and lengths that are proportional.
All the lengths in B are multiplied by a scale factor, \(k\).

We are given that one length, the diagonal, is scaled up by factor \(k=3\)

Volume = Length * Length * Length

So an increase in volume from B to A equals each of those three lengths * \(k\).
That is, change in volume = \(k^3\)

\(k^3=3^3=27\)

(Sidebar: Area is Length * Length, so multiply smaller area by \(k^2\). Increase in just one length is (Length)* \(k\))

asma , you are correct that if we use "side" as a basis, the diagonal of a face of a cube is
(side of square) * \(\sqrt{2}\)


PKN , did you mean \(\sqrt{2}\)?
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The diagonal of one face of cube P is three times as long as the diago 11 Sep 2018, 19:16
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AsmaM
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Bunuel
The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64

Diagonal of a cube=\(\sqrt{3}\)*side length

Given, the diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q.
Or, \(\sqrt{3}\)*side length of cube P=3*\(\sqrt{3}\)*side length of cube Q
Or, side length of cube P=3*side length of cube Q

Volume of cube Q=\((1/3*Side-length-of-cube-P)^3\)=\(\frac{1}{27}\)* Volume of cube P

Ans. (D)
Why is the diagonal of a face of a cube \(\sqrt{3}\)*side length and not \(\sqrt{2}\)*side length according to our 45-45-90 right triangle?

It's the same answer in the end, though
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Hi AsmaM generis,
It was a typo indeed that need to be taken care at my end, athough answer remains the same.
Face diagonal of a cube with side length 'a' is \(a\sqrt{2}\) since each face of a cube is a SQUARE.

Thank you
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The diagonal of one face of cube P is three times as long as the diago 14 Sep 2018, 17:43
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Bunuel
The diagonal of one face of cube P is three times as long as the diagonal of one face of cube Q. To find the volume of cube Q one should divide the volume of cube P by


A. \(2\sqrt{2}\)

B. 3

C. 8

D. 27

E. 64
Show more


Let’s let P = the length of a side of cube P and Q = the length of a side of cube Q. We see that the diagonal of a face of cube P is P√2, and the diagonal of a face of cube Q is Q√2. We can create the equation:

P√2 = 3(Q√2)

P = 3Q

P^3 = 27Q^3

P^3/27 = Q^3

Answer: D
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The diagonal of one face of cube P is three times as long as the diago 28 Mar 2024, 19:56
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