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# What is the volume of a certain rectangular solid?

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Re: OG-12 DS # 122 [#permalink]

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16 Jan 2012, 21:07
From the question, we do not know if the rectangular solid is a cube or a cuboid.

Fact 1 - Two adjacent faces of the solid have areas 15 and 24, respectively => This implies that the rectangular solid is a cuboid. We need to know the length of all 3 sides to calculate the volume. From the 2 adjacent face areas (15 & 24), we do not exactly know the length of the sides. If the sides are a, b & c => two adjacent faces could be made up of (a,b) and (b,c) OR (a,c) and (b,c).

Either way we cannot conclusively calculate all the 3 sides. (BCE)

Fact 2 - Each of the two opposite faces of the solid has area 40. Firstly this is infact a little confusing. I go as far as to think that all the faces have an area of 40. But I strongly believe in GMAC's ability to NOT confuse people and only give relevant information. So since the first fact dealt with adjacent faces, the 2nd fact deals with opposite faces. Opposite faces of a rectangular solid MUST have the same area but with this fact, we do not know which face we are talking about - just that one of the faces has an area of 40. And also that its opposite face is also 40 (which goes without saying!) And we still do not know the length of the sides that make up this face lest all the 3 sides. (1 X 40 = 40 OR 5 X 8 = 40). So this too is insufficient. (CE)

Combining both statements => we now know all the 3 face areas. 15, 24, and 40.

Say sides are a, b, c then

ab=15
bc=24
ac=40

Volume = abc
Multiply all 3 values ab X bc X ac = (abc)^2 = (15 x 24 x 40) = whatever it is, we can calculate the value of abc, the volume.

I went back to the forums to confirm my method but yes most of them agree that the 2nd fact is confusing. And the explanation in the OG for the 2nd fact is that "...the volume is (5)(8)(x), which will vary as x varies.." - I think this is their way of saying that we only know the area of one face.

Hope this helps!
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Re: OG-12 DS # 122 [#permalink]

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29 Apr 2012, 03:39
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

hey, can anyone describe this last step? whats the rule for that or where does it come from?

Thanks
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Re: OG-12 DS # 122 [#permalink]

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29 Apr 2012, 05:43
andih wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

hey, can anyone describe this last step? whats the rule for that or where does it come from?

Thanks

Do you mean multiplying?

If a=2, b=3, and c=4 then abc=2*3*4=24.
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Re: What is the volume of a certain rectangular solid? [#permalink]

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22 Jul 2013, 14:00
I am confused. Why can't we solve the volume with (1)?

Area of 2 sides is given: 15 and 24. That means they share a side and shouldn't the side be 3?

15: {1,15}, {3,5}
24: {1,24}, {2,12}, {3,8}, {4,6}

therefore the sides are 3, 5, and 8 and the volume is 3x5x8=120
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Re: What is the volume of a certain rectangular solid? [#permalink]

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22 Jul 2013, 14:03
bugzor wrote:
I am confused. Why can't we solve the volume with (1)?

Area of 2 sides is given: 15 and 24. That means they share a side and shouldn't the side be 3?

15: {1,15}, {3,5}
24: {1,24}, {2,12}, {3,8}, {4,6}

therefore the sides are 3, 5, and 8 and the volume is 3x5x8=120

Please check here: what-is-the-volume-of-a-certain-rectangular-solid-90748.html#p772095 Notice that there are examples given showing that we can have more than one answer.

Hope it helps.
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Re: What is the volume of a certain rectangular solid? [#permalink]

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22 Jul 2013, 14:05
Bunuel wrote:
bugzor wrote:
I am confused. Why can't we solve the volume with (1)?

Area of 2 sides is given: 15 and 24. That means they share a side and shouldn't the side be 3?

15: {1,15}, {3,5}
24: {1,24}, {2,12}, {3,8}, {4,6}

therefore the sides are 3, 5, and 8 and the volume is 3x5x8=120

Please check here: what-is-the-volume-of-a-certain-rectangular-solid-90748.html#p772095 Notice that there are examples given showing that we can have more than one answer.

Hope it helps.

Oh! I totally overlooked side=1. thanks
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Re: OG-12 DS # 122 [#permalink]

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10 Dec 2013, 09:35
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Wouldn't the equation be (15*40*24)^2?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

Last edited by WholeLottaLove on 10 Dec 2013, 09:41, edited 1 time in total.
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Re: OG-12 DS # 122 [#permalink]

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10 Dec 2013, 09:41
WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

When we combine the statements we have:
$$blue=d*h=15$$.

$$yellow=l*h=24$$.

$$red=d*l=40$$.

Multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it's clear.
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Re: OG-12 DS # 122 [#permalink]

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10 Dec 2013, 09:45
Hello,

I tried to edit my question in time to better specify my question. I see that in 1 and 2 there are two equations with three unknowns whereas here, we have equations for length, depth, height. You multiply 15*40*24 but according to your formula, don't you multiply them together then square that result? Then how do you go from 24^2*5^2 to 24*5? I see that you take the root but why is that done and how in the context of this problem?

Thanks!

Bunuel wrote:
WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

When we combine the statements we have:
$$blue=d*h=15$$
$$yellow=l*h=24$$
$$red=d*l=40$$

Multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it's clear.
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Re: OG-12 DS # 122 [#permalink]

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10 Dec 2013, 09:52
WholeLottaLove wrote:
Hello,

I tried to edit my question in time to better specify my question. I see that in 1 and 2 there are two equations with three unknowns whereas here, we have equations for length, depth, height. You multiply 15*40*24 but according to your formula, don't you multiply them together then square that result? Then how do you go from 24^2*5^2 to 24*5? I see that you take the root but why is that done and how in the context of this problem?

Thanks!

Bunuel wrote:
WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?

When we combine the statements we have:
$$blue=d*h=15$$
$$yellow=l*h=24$$
$$red=d*l=40$$

Multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it's clear.

$$Volume=l*h*d$$.

Now, if you multiply the 3 equations we have we get $$l^2*h^2*d^2*=(lhd)^2=15*24*40=24^2*5^2$$ --> $$Volume=l*h*d$$, thus the volume is the square root of $$(lhd)^2=24^2*5^2$$, so 24*5=120.

Hope it's clear.
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Re: What is the volume of a certain rectangular solid? [#permalink]

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01 Feb 2014, 19:49
2
KUDOS
This is more like a verbal question on meaning. Trick here is statement2 which makes this question hard to understand and easy to get it wrong.

(2) Each of two opposite faces of the solid has area 40.
===>someone can interpret this as pick any 2 opposite sides the area is 40. or exactly 2 opposite sides have areas of 40 each. but if you look at statement one then the latter makes more sense, thats the trick. Since OG question do not contradict on Statement1 and 2, this is an example of how you can verify the statement more carefully just by knowing what happened in 1. But remember while evaluating 2 do not bring that concept in 1. that seems hard to do in this question.

so for me this question is dubious, although answer should be C.
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Re: OG-12 DS # 122 [#permalink]

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18 May 2014, 06:50
Bunuel wrote:
mainhoon wrote:
If I refer to your drawing above. Blue face=15 Yellow face=24 and Red face=40. Bunuel, may be its just the wording that I don't understand:

"(1) Two adjacent faces of the solid have areas 15 and 24, respectively" means two faces have areas 15 and 24. We could say it ourselves that there will be two faces with these areas which are adjacent. How else?

>> I took this statement to mean this: Two adjacent faces could refer to: Red+Blue, Red+Yellow, Blue+Yellow. The assumption is that all faces are treated equal. Clearly, not all adjacent faces have area 15,24.

"(2) Each of two opposite faces of the solid has area 40" means that one pair of opposite faces has an area 40.

>> Each of two opposite faces = 40 to me means any opposite faces =40. I can understand if it said one pair, but I am not sure if "each of two" means "one pair". Clearly not all opposite faces have area = 40.

Thanks for your time and help.

Yes you understanding of wording is wrong. The question means exactly what I wrote.

Hi Bunuel

On hindsight and after seeing the OG , we can say that the answer is C but the option does sound a confusing .. Shouldn't it have been ' " One of the two opposite faces of the solid has area = 40." By qualifying as each , doesn't it cover each of the 3 pair of faces thus all 6 faces ?
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Re: What is the volume of a certain rectangular solid? [#permalink]

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04 Jun 2014, 03:00
Hi bunuel,
Though your explanation is correct but st#2 is confusing
It says:
Each of two opposite faces of the solid has area 40.

"Each" has area 40. It means from your fig, Red will have area =40 , but Blue face and (opposite to it face) both can also have area 40..

Thats how it is confusing. Can you clarify
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Re: What is the volume of a certain rectangular solid? [#permalink]

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04 Jun 2014, 03:12
nidhi12 wrote:
Hi bunuel,
Though your explanation is correct but st#2 is confusing
It says:
Each of two opposite faces of the solid has area 40.

"Each" has area 40. It means from your fig, Red will have area =40 , but Blue face and (opposite to it face) both can also have area 40..

Thats how it is confusing. Can you clarify

Does it matter which two opposite faces have the area 40?

"Each of two opposite faces of the solid has area 40" means that one pair of opposite faces has area 40.
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Re: What is the volume of a certain rectangular solid? [#permalink]

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21 Jun 2014, 18:21
tingle15 wrote:
Yes, the second statement is confusing but the catch is in the question itself. The question states a rectangular solid. All the faces cannot have an area of 40 if the solid is rectangular.

Well explained. But yes, the second statement is very confusing. I get it now though. But it's intentionally tricky, or just poorly worded.
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What is the volume of a certain rectangular solid? [#permalink]

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30 Oct 2014, 10:05
how is a cube not a rectangular solid?
why should we look at statement 1 when analyzing statement 2?
from what I know, we do not have to confuse between statements when analyzing only one statement!!!
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Re: What is the volume of a certain rectangular solid? [#permalink]

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01 Nov 2014, 04:27
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

In question, 2nd statement is confusing. It says "Each of two opposite faces of the solid has area 40". I interpreted it wrong. I understood it as all opposite faces are 40.
so (length*width)(width*height)(length*height) = 40*40*40
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Re: What is the volume of a certain rectangular solid? [#permalink]

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01 Nov 2014, 04:27
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

In question, 2nd statement is confusing. It says "Each of two opposite faces of the solid has area 40". I interpreted it wrong. I understood it as all opposite faces are 40.
so (length*width)(width*height)(length*height) = 40*40*40
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Re: What is the volume of a certain rectangular solid? [#permalink]

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01 Nov 2014, 05:48
ammuseeru wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> $$blue=d*h=15$$ and $$yellow=l*h=24$$ --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns ($$V=l*h*d$$).

To elaborate more:
If $$blue=d*h=15*1=15$$ and $$yellow=l*h=24*1=24$$ then $$V=l*h*d=24*1*15=360$$;
If $$blue=d*h=5*3=15$$ and $$yellow=l*h=8*3=24$$ then $$V=l*h*d=8*3*5=90$$.

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): $$blue=d*h=15$$, $$yellow=l*h=24$$ and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: $$red=d*l=40$$ --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate $$V=l*h*d$$. Sufficient.

To show how it can be done: multiply these 3 equations --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

In question, 2nd statement is confusing. It says "Each of two opposite faces of the solid has area 40". I interpreted it wrong. I understood it as all opposite faces are 40.
so (length*width)(width*height)(length*height) = 40*40*40

indeed! in a rectangular solid, there are 3 opposite faces. Here the statement says EACH!!! the statement is very ambiguous.
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Re: What is the volume of a certain rectangular solid? [#permalink]

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08 Dec 2014, 06:45
deepbidwai wrote:
Information in (2) is redundant as from (1) we can know all required info.
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
So, one common side has to be 3. i.e. 5*3 and 8*3. From this info we can say area of the face is 8*5= 40.
Hence only (1) is sufficient.
Answer should be A and not C.

DeepakB

I also don't understand why A is not sufficient.

We need to determine the length of 3 sides.

If two adjacent faces share a side with respective areas 15 and 24, the only common prime they share is 3, which must be their shared side length.

If the area of one face is 15 and one side is 3, then the other side is 5.
If the area of one face is 24 and one side is 3, then the other side is 8.

Therefore, we have three side lengths: 5, 8, 3

Perhaps my mistake is assuming that the dimensions must be integers?
Re: What is the volume of a certain rectangular solid?   [#permalink] 08 Dec 2014, 06:45

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