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What is the volume of a certain rectangular solid? [#permalink]

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31 Aug 2010, 06:26

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This question is from Official Guide and Official Answer is C.

About rectangular solid:

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 --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Bunuel, Can you clarify if 1 and 2 conflict in this case?

There is no contradiction. It is clearly mentioned each of the two opposite faces. That means the question is referring 2 opposite faces only. READ CAREFULLY

If the area is 40 then 5*8 = 40 along with the 3*5 and 3*8 of the 1st statement is helpful in uniquely identifying the volume 3*5*8 as asked by the question.

Else area 15 and 24 could be because of the sides 1,15,24. The second statement removes this alternate solution. _________________

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

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01 Feb 2014, 19:49

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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. _________________

What is the volume of a certain rectangular solid? (1) Two adjacent faces of the solid have areas 15 and 24, respectively. (2) Each of two opposite faces of the solid has area 40.

Do you understand very well the second statement? I understand that if you pick ANY two faces of the solid, each of them has area 40.

Let the sides be a,b,c

(1) ab=15 & bc=24. Not enough to calculate abc which is the volume. Insufficient

(2) Do you understand very well the second statement? Yes, its clear, it means that if you pick any of the two, it will have area=40. Not enough to calculate volume, just one area provided

(1+2) Now we know all three areas, ab, bc, ca as 15,24,40 If you multiply all three, \(a^2b^2c^2 = 15x24x40\). Just take the square root to get the volume (abc). Sufficient !!

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.

1) Two adjacent faces of the slid have areas 15 & 24, respectively.

2) Each of the opposite faces of the solid has area 40.

Let the length of the sides be a,b,c. Areas of each of the sides will be ab,bc,ca and the volume is abc

1. ab=15 ac=24 From these we cannot get individual values of a,b,c or the volume. So INSUFFICIENT

2. bc=40. From this we cannot get the volume. -- INSUFFICIENT

Combining we have:

ab=15 ac=24 bc=40

multiplying the LHS and RHS, you get:

(ab)(ac)(bc)=15*24*40 (a^2)(b^2)(c^2)=14400 or abc=120=Volume

So, C is the answer

How does 2nd statement imply that only bc=40, the statement says that EACH of the opposite faces of the solid has area 40, so by this , even ac=ab=40, right? _________________

Working without expecting fruit helps in mastering the art of doing fault-free action !

1) Two adjacent faces of the slid have areas 15 & 24, respectively.

2) Each of the opposite faces of the solid has area 40.

Let the length of the sides be a,b,c. Areas of each of the sides will be ab,bc,ca and the volume is abc

1. ab=15 ac=24 From these we cannot get individual values of a,b,c or the volume. So INSUFFICIENT

2. bc=40. From this we cannot get the volume. -- INSUFFICIENT

Combining we have:

ab=15 ac=24 bc=40

multiplying the LHS and RHS, you get:

(ab)(ac)(bc)=15*24*40 (a^2)(b^2)(c^2)=14400 or abc=120=Volume

So, C is the answer

How does 2nd statement imply that only bc=40, the statement says that EACH of the opposite faces of the solid has area 40, so by this , even ac=ab=40, right?

Since we have the area of two adj sides is it not possible to find the length of the side that these two sides have in common and from there find the volume. Ie 15: 15, 5, 3, 1 and 24: 24, 12, 8, 6, 4, 3, 2, 1. The number three is the only one in common so this would be either the common base or height and we can go from there...please advise. Thanks!

I think that two statements contradict each other, because: Statement 2 states that each face has area 40, but st. 1 says that some faces are 15 and 24 each... (I assume that face=side)

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.

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

from (1)

ab =15 and bc = 24 OR ab = 15 and ac = 24

therefore you have no idea how to find the third side

if you think about it logicaly, you have three unknowns, therefore you will need three equations

Hi gurpreetsingh Thanks for the input. Well I see your point, however here is my dilemma even with sides of 5,3,8. stm1 - adjacent areas are 15, 24 ok. stm2 - each of the opposite faces has area 40. now each of the opposite face can also be one of the faces referred to in stmt1 and that does not have area of 40. in fact i could look at the area40 face and consider it to be one of the adjacent areas referred to in stm1 and then it is no longer 15 or 24 in area. I assume that all faces are treated equally in the solid, so how do I differentiate what is opposite and what is adjacent? _________________

This question is from Official Guide and Official Answer is C.

About rectangular solid:

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 --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.

But Bunuel, what I don't follow is this. As you had said statements cannot contradict each other. statement 2: Each of the opposite faces is area 40. This means if I pick a side of the solid you have drawn above then that and the side opposite to it will each have area 40. Clearly that is not the case, it is only true for one pair of the sides. Same argument with statement 1: adjacent faces are 15 and 24, if I picked one of the adjacent areas to be 40 then that is not true either. So while I understand the solution, I would like to know where am I wrong. Thanks _________________

But Bunuel, what I don't follow is this. As you had said statements cannot contradict each other. statement 2: Each of the opposite faces is area 40. This means if I pick a side of the solid you have drawn above then that and the side opposite to it will each have area 40. Clearly that is not the case, it is only true for one pair of the sides. Same argument with statement 1: adjacent faces are 15 and 24, if I picked one of the adjacent areas to be 40 then that is not true either. So while I understand the solution, I would like to know where am I wrong. Thanks

Not sure I understand your question.

"(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?

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

Now, when combining: there will be 3 faces of the solid with given areas which are adjacent. How else? _________________

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.

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