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The diagonal of one face of cube P is three times as long as the diago

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The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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09 Sep 2018, 06:58
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25% (medium)

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77% (01:34) correct 23% (01:12) wrong based on 40 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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Re: The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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09 Sep 2018, 08:27
Bunuel wrote:
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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The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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11 Sep 2018, 05:30
PKN wrote:
Bunuel wrote:
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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The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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11 Sep 2018, 17:50
Bunuel wrote:
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

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 = $$1^3=1$$

$$\frac{Vol_{A}}{vol_{B}}=\frac{27}{1}=27$$

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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Re: The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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11 Sep 2018, 18:16
AsmaM wrote:
PKN wrote:
Bunuel wrote:
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

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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PKN

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Re: The diagonal of one face of cube P is three times as long as the diago  [#permalink]

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14 Sep 2018, 16:43
Bunuel wrote:
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

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

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Re: The diagonal of one face of cube P is three times as long as the diago   [#permalink] 14 Sep 2018, 16:43
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